Duality Theory in Fuzzy Linear Programming Problems with Fuzzy Coefficients

The concept of fuzzy scalar (inner) product that will be used in the fuzzy objective and inequality constraints of the fuzzy primal and dual linear programming problems with fuzzy coefficients is proposed in this paper. We also introduce a solution concept that is essentially similar to the notion of Pareto optimal solution in the multiobjective programming problems by imposing a partial ordering on the set of all fuzzy numbers. We then prove the weak and strong duality theorems for fuzzy linear programming problems with fuzzy coefficients.     

The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions

The optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts for these kinds of problems will follow the concept of nondominated solution adopted in the multiobjective programming problems. In order to consider the differentiation of fuzzy-valued functions, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the (strongly, weakly) Pareto optimal solutions are elicited naturally by introducing the Lagrange multipliers.     

Ag8Cl2[Se2P(OEt)2]6: A rare example containing a combination of discrete clusters and chains

The novel halide-centered Ag(I)8 cubic clusters containing diethyl diselenophosphato ligands are prepared and their solid state structures, a discrete unit or a one-dimensional chain, are dictated by the counter anions.     

Fuzzy-valued integrals based on a constructive methodology

The procedures for constructing a fuzzy number and a fuzzy-valued function from a family of closed intervals and two families of real-valued functions, respectively, are proposed in this paper. The constructive methodology follows from the form of the well-known “Resolution Identity” (decomposition theorem) in fuzzy sets theory. The fuzzy-valued measure is also proposed by introducing the notion of convergence for a sequence of fuzzy numbers. Under this setting, we develop the fuzzy-valued integral of fuzzy-valued function with respect to fuzzy-valued measure. Finally, we provide a Dominated Convergence Theorem for fuzzy-valued integrals.     

Saddle Point Optimality Conditions in Fuzzy Optimization Problems

The fuzzy-valued Lagrangian function of fuzzy optimization problem via the concept of fuzzy scalar (inner) product is proposed. A solution concept of fuzzy optimization problem, which is essentially similar to the notion of Pareto solution in multiobjective optimization problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. Under these settings, the saddle point optimality conditions along with necessary and sufficient conditions for the absence of a duality gap are elicited.     

Hahn-Banach theorems in nonstandard normed interval spaces

The conventional Hahn-Banach extension theorem based on vector space has been widely used to obtain many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the interval space is not a real vector space, the Hahn-Banach extension theorems based on interval spaces and nonstandard normed interval spaces can still be derived in this paper, which also shows the possible applications by considering the interval-valued problems in nonlinear analysis, vector optimization and mathematical economics.     

The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function

The KKT conditions in an optimization problem with interval-valued objective function are derived in this paper. Two solution concepts of this optimization problem are proposed by considering two partial orderings on the set of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we derive the KKT optimality conditions.