Mehar’s method for solving fully fuzzy linear programming problems with L-R fuzzy parameters

To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers.In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods.     

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.     

A new predictor–corrector method for solving unconstrained minimization problems

This paper presents a new predictor–corrector method for finding a local minimum of a twice continuously differentiable function. The method successively constructs an approximation to the solution curve and determines a predictor on it using a technique similar to that used in trust region methods for unconstrained optimization. The proposed predictor is expected to be more effective than Euler's predictor in the sense that the former is usually much closer to the solution curve than the latter for the same step size. Results of numerical experiments are reported to demonstrate the effectiveness of the proposed method.     

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...     

A “build-down” scheme for linear programming

We propose a build-down scheme for Karmarkar's algorithm and the simplex method for linear programming. The scheme starts with an optimal basis candidate set including all columns of the constraint matrix, then constructs a dual ellipsoid containing all optimal dual solutions. A pricing rule is developed for checking whether or not a dual hyperplane corresponding to a column intersects the containing ellipsoid. If the dual hyperplane has no intersection with the ellipsoid, its corresponding column will not appear in any of the optimal bases, and can be eliminated from. As these methods iterate, is eventually built-down to a set that contains only the optimal basic columns.     

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.     

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.     

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.     

A note on noncommutative moment problems

Noncommutative moment problems for C*-algebras are studied. We generalize a result of Hadwin on tracial states to nontracial case. Our results are applied to obtain simple solutions to moment problems on the square and the circle as well as extend the positive unital functionals from a (discrete) complex group algebra to states on the group C*-algebra.     

A Barzilai–Borwein type method for stochastic linear complementarity problems

We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.     

A note on the Douglas–Rachford splitting method for optimization problems involving hypoconvex functions

Recently, the convergence of the Douglas–Rachford splitting method (DRSM) was established for minimizing the sum of a nonsmooth strongly convex function and a nonsmooth hypoconvex function under the assumption that the strong convexity constant \(\beta \) is larger than the hypoconvexity constant \(\omega \). Such an assumption, implying the strong convexity of the objective function, precludes many interesting applications. In this paper, we prove the convergence of the DRSM for the case \(\beta =\omega \), under relatively mild assumptions compared with some existing work in the literature.     

A quadratic finite element method for solving biharmonic problems in ℝ n

Summary A family of simplicial finite element methods having the simplest possible structure, is introduced to solve biharmonic problems in ⁿ ,n3, using the primal variable. The family is inspired in the MORLEY triangle for the two dimensional case, and in some sense this element can be viewed as its member corresponding to the valuen=2.     

A generalized Laguerre–Legendre spectral collocation method for solving initial-boundary value problems

In this research, a mixed spectral collocation method based on Kronecker product is proposed for solving initial-boundary value problems. New implementation is suggested to achieve more accurate approximation at longer times. Test problems are also studied to demonstrate how this method is implemented. Numerical experiments reveal that the new method is very effective and convenient.     

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.     

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.     

A modification of the method of linearization for solving a mathematical programming problem on a simple set of “parallelepiped” type

We propose a modification of B. N. Pshenichnyi's method of linearization for solving mathematical programming problems on a simple set of parallelepiped type. We study in detail the computational aspects and the efficiency of the approach in question. Two tables. Bibliography: 3 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 24–28.