Multiparametric Linear Programming Research Articles

Linear Multiparametric Programming by Multicriteria Simplex Method

A recently derived Multicriteria Simplex Method [Yu, P. L., M. Zeleny. 1975. The set of all nondominated solutions in linear cases and a multicriteria simplex method. J. Math. Anal. Appl. 49 (2, February) 430–468.] is used to study some basic properties in the decomposition of parametric space. A new type of parametric space, which arises naturally in its formulation, is used. Two computational methods are discussed. The first one is an indirect algebraic method which locates the corresponding set of all nondominated extreme points. The second one is a direct geometric decomposition method which is similar to that discussed by Gal and Nedoma [Gal, T., J. Nedoma. 1972 Multiparametric linear programming. Management Sci.. 18 406–421.]. The difficulties of such a geometric method are discussed through an example.

Rim Multiparametric Linear Programming

The rim multiparametric linear programming problem (RMPLP) is a parametric problem with a vector-parameter in both the right-hand side and objective function (i.e., in the “rim”). The RMPLP determines the region K* ⊂ E* such that the problem, maximize z(λ) = cT(λ)x, subject to Ax = b(λ), x ≧ 0, has a finite optimal solution for all λ ∈ K*. Let Bi be an optimal basis to the given problem, and let Ri*, be a region assigned to Bi such that for all λ ∈ Ri* the basis Bi is optimal. The goal of the RMPLP problem is to cover K* by the Ri* such that the various Ri* do not overlap. The purpose of this paper is to present a solution method for finding all regions Ri* that cover K* and do not overlap. This method is based upon an algorithm for a multiparametric problem described in an earlier paper by Gal and Nedoma.

Multiparametric Linear Programming

The multiparametric linear programming (MLP) problem for the right-hand sides (RHS) is to maximize z = cTx subject to Ax = b(λ), x ≧ 0, where b(λ) be expressed in the form [Formula: see text] where F is a matrix of constant coefficients, and λ is a vector-parameter. The multiparametric linear programming (MLP) problem for the prices or objective function coefficients (OFC) is to maximize z = cT(v)x subject to Ax = b, x ≧ 0, where c(I) can be expressed in the form c(v) = c* + Hv, and where H is a matrix of constant coefficients, and v a vector-parameter. Let Bi be an optimal basis to the MLP-RHS problem and Ri be a region assigned to Bi such that for all λ ϵ Ri the basis Bi is optimal. Let K denote a region such that K = UiRi provided that the Ri for various I do not overlap. The purpose of this paper is to present an effective method for finding all regions Ri that cover K and do not overlap. This method uses an algorithm that finds all nodes of a finite connected graph. This method uses an algorithm that finds all nodes of a finite connected graph. An analogus method is presented for the MLP-OFC problem.

Probleme der linearen optimierung mit nichtilinear-einparametrischen koeffizienten in der zielfunktion

Linear programming problems with coefficients of the objective function, depending non linear from one parameter, may be solved using the theory of multiparametric linear programming (cf. Sokolova [3]). The condition under which this theory can be used is a special kind of dependence for each coefficient that consists practically in representing each coefficient as a linear combination with constant and variable factors with a known variation (e.g. whether, variation of different economic factors etc.). One gets subsets of the set on which the curve appearing in such a manner in the parameter space is defined by intersecting this curve with certain maximal polyhedrons, arising by computing the associated multiparametric problem. All elements of such a subset have either the same optimal solution or an unbounded objective function. At least one gives a computing procedure for the described problems.