A note on “A new method for solving fully fuzzy linear programming problems”

This note shows that solving fully fuzzy linear programming (FFLP) model presented by Kumar et al. [A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Model. 35 (2011) 817–823] needs some corrections to make the model well in general. A new version is provided in this note. A simple example is also presented to demonstrate the new form.     

Mehar’s methods for fuzzy assignment problems with restrictions

In this paper, limitations of existing methods [5, 11] for solving fuzzy assignment problems (FAPs) are pointed out. In order to overcome the limitations of existing methods, two new methods named Mehar’s methods are proposed. To show the advantages of Mehar’s methods over existing methods, some FAPs are solved. The Mehar’s methods can solve the problems solved by existing methods as well as those which cannot be solved by existing methods.     

Mehar’s method to find exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems

To the best of our knowledge, till now there is no method described in literature to find exact fuzzy optimal solution of balanced as well as unbalanced fully fuzzy multi-objective transportation problems. In this paper, a new method named as Mehar??s method, is proposed to find the exact fuzzy optimal solution of fully fuzzy multi-objective transportation problems (FFMOTP). The advantages of the Mehar??s method over existing methods are also discussed. To show the advantages of the proposed method over existing methods, some FFMOTP, which cannot be solved by using any of the existing methods, are solved by using the proposed method and the results obtained are discussed. To illustrate the applicability of the Mehar??s method, a real life problem is solved.     

Comment on “Duality theory in fuzzy linear programming problems with fuzzy coefficients”

This note provides a counterexample to illustrate the incorrectness of the proof of Proposition 3.3 that was presented by Wu (Fuzzy Optim Decis Mak 2:61–73, 2003). The original proof of Proposition 3.3 by Wu can only be correct when the extra assumption \(\mu _{\widetilde{y}_i}(0)= 1\) is added. The correct proof of Proposition 3.3 is also presented in this note.     

The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

In this paper, we present an efficient numerical algorithm for solving a general class of nonlinear singular boundary value problems. This present algorithm is based on the Adomian decomposition method (ADM) and Green’s function. The method depends on constructing Green’s function before establishing the recursive scheme. In contrast to the existing recursive schemes based on ADM, the proposed numerical algorithm avoids solving a sequence of transcendental equations for the undetermined coefficients. The approximate series solution is calculated in the form of series with easily computable components. Moreover, the convergence analysis and error estimation of the proposed method is given. Furthermore, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results reveal that the proposed method is very effective.     

High-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion

An approach to the construction of high-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion is proposed. Godunov’s theorem on monotone schemes is used to construct a new antidiffusion flux limiter in high-order accurate difference schemes as applied to linear advection equations with constant coefficients. The efficiency of the approach is demonstrated by solving linear advection equations with constant coefficients and one-dimensional gasdynamic equations.     

Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials

In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving the twelfth-order boundary-value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Regularization method with two parameters for nonlinear ill-posed problems

This paper is devoted to the regularization of a class of nonlinear ill-posed problems in Banach spaces. The operators involved are multi-valued and the data are assumed to be known approximately. Under the assumption that the original problem is solvable, a strongly convergent approximation procedure is designed by means of the Tikhonov regularization method with two pa- rameters.     

An interactive fuzzy satisficing method for multiobjective 0–1 programming problems with fuzzy numbers through genetic algorithms with double strings

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem-formulation process, multiobjective 0–1 programming problems involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy α-programming problem is introduced. The fuzzy goals of the decision maker (DM) for the objective functions are quantified by eliciting the corresponding linear membership functions. Through the introduction of an extended Pareto optimality concept, if the DM specifies the degree α and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the augmented minimax problems through genetic algorithms with double strings. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the DM efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.     

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.     

Some new perspectives for solving 0–1 integer programming problems using Balas method

Computational Management Science - Egon Balas’s additive algorithm, also known as implicit enumeration, is a technique that uses a branch-and-bound (B&B) approach to finding optimal...     

On solving systems of linear algebraic equations by Gibbs’s method

The paper presents an algorithm of approximate solution of a system of linear algebraic equations by the Monte Carlo method superimposed with ideas of simulating Gibbs and Metropolis fields. A solution in the form of a Neumann series is evaluated, the whole vector of solutions is obtained. The dimension of a system may be quite large. Formulas for evaluating the covariance matrix of a single simulation run are given. The method of solution is conceptually linked to the method put forward in a 2009 paper by Ermakov and Rukavishnikova. Examples of 3 × 3 and 100 × 100 systems are considered to compare the accuracy of approximation for the method proposed, for Ermakov and Rukavishnikova’s method and for the classical Monte Carlo method, which consists in consecutive estimation of the components of an unknown vector.     

On solving the problems of stability by Lyapunov’s direct method

We consider problems of stability and instability of the trivial solution to nonautonomous systems of differential equations. We suggest new theorems of Lyapunov’s direct method with the use of semi-definite auxiliary functions. The idea is based on the use of the additional function that evaluates the rate of convergence of the solutions to the set, where Lyapunov’s function vanishes. We formulate theorems on the non-asymptotic stability and instability. The results are illustrated by examples, where we give a comparison with known results.     

Interactive fuzzy programming for multi-level 0–1 programming problems through genetic algorithms

In this paper, we propose interactive fuzzy programming for multi-level 0–1 programming problems through genetic algorithms. Our method is supposed to apply to hierarchical decision problems in which decision-making at each level is sequential from upper to lower level and decision makers are essentially cooperative. After determining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived efficiently by updating the satisfactory degrees of the decision makers at the upper level with considerations of overall satisfactory balance among all levels. An illustrative numerical example for three-level 0–1 programming problems is provided to demonstrate the feasibility of the proposed method.     

Barzilai and Borwein’s method for multiobjective optimization problems

The present study is an attempt to extend Barzilai and Borwein’s method for dealing with unconstrained single objective optimization problems to multiobjective ones. As compared with Newton, Quasi-Newton and steepest descent multi-objective optimization methods, Barzilai and Borwein multiobjective optimization (BBMO) method requires simple and quick calculations in that it makes no use of the line search methods like the Armijo rule that necessitates function evaluations at each iteration. It goes without saying that the innovative aspect of the current study is due to the use of no function evaluations in comparison with other multi-objective optimization non-parametric methods (e.g. Newton, Quasi-Newton and steepest descent methods, to name a few) that have been investigated so far. Also, the convergence of the BBMO method for the objective functions assumed to be twice continuously differentiable has been proved. MATLAB software was utilized to implement the BBMO method, and the results were compared with the other methods mentioned earlier. Using some performance assessment, the quality of nondominated frontier of BBMO was analogized to above mentioned methods. In addition, the approximate nondominated frontiers gained from the methods were compared with the exact nondominated frontier for some problems. Also, performance profiles are considered to visualize numerical results presented in tables.     

Barzilai–Borwein method with variable sample size for stochastic linear complementarity problems

A smoothing method for solving stochastic linear complementarity problems is proposed. The expected residual minimization reformulation of the problem is considered, and it is approximated by the sample average approximation (SAA). The proposed method is based on sequential solving of a sequence of smoothing problems where each of the smoothing problems is defined with its own sample average approximation. A nonmonotone line search with a variant of the Barzilai–Borwein (BB) gradient direction is used for solving each of the smoothing problems. The BB search direction is efficient and low cost, particularly suitable for nonmonotone line search procedure. The variable sample size scheme allows the sample size to vary across the iterations and the method tends to use smaller sample size far away from the solution. The key point of this strategy is a good balance between the variable sample size strategy, the smoothing sequence and nonmonotonicity. Eventually, the maximal sample size is used and the SAA problem is solved. Presented numerical results indicate that the proposed strategy reduces the overall computational cost.     

On Schweiger’s problems on fully subtractive algorithms

In his monograph [6] on multi-dimensional continued fractions, Schweiger has presented two conjectures on fully subtractive algorithms. We affirm one and refute another.     

A Douglas–Rachford splitting method for solving equilibrium problems

We propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition. Our algorithm is a consequence of the Douglas–Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.     

Korpelevich’s method for variational inequality problems in Banach spaces

We propose a variant of Korpelevich’s method for solving variational inequality problems with operators in Banach spaces. A full convergence analysis of the method is presented under reasonable assumptions on the problem data.