In this paper, the notions of $(T,S)$-composition matrix and$(T,S)$-interval-valued intuitionistic fuzzy equivalence matrix areintroduced where $(T,S)$ is a dual pair of triangular module. Theyare the generalization of composition matrix and interval-valuedintuitionistic fuzzy equivalence matrix. Furthermore, theirproperties and characterizations are presented. Then a new methodbased on $tilde{...

Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. We consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problem...

We describe primal, dual and primal-dual steepest-descent potential reduction methods for linear and convex minimization on the simplex in Rn and present their precise iteration complexity analyses depending on the problem dimension and the Lipschitz parameter of the constraint matrix or the convex objective function. In these methods, no matrix needs to be ever inversed so that they are pure f...

This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming (LP) problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple Forrest-Tomlin upda...

Maximize ZD = b T y s.t. (D) A y + s = c, s ≥ 0n. where we have here chosen to add slack variables s ≥ 0n. The simplex method, as presented previously, works by starting with an initial basic feasible solution to (P) (which can be found via a phase 1 problem). However, what if a dual feasible basis is immediately available, whereas a basic feasible solution to (P) is harder to obtain? Can this ...

The dual Phase-I algorithm using the most-obtuse-angle row pivot rule is very efficient for providing a dual feasible basis, in either the classical or the basisdeficiency-allowing context. In this paper, we establish a basis-deficiency-allowing Phase-I algorithm using the so-called most-obtuse-angle column pivot rule to produce a primal (deficient or full) basis. Our computational experiments ...