Partial stability of multi attribute decision-making solutions for interval determined criteria weights - the problem of nonlinear programming
Abstract
Introduction/purpose: The paper presents a designed procedure for solving a class of nonlinear programming (NLP) tasks with the nonlinear and differentiable objective function, linear natural constraints (intervals of possible arguments values - variables) and the normalization condition for arguments. The procedure was applied to determine the partial stability of the solution of the problem of multi attibute decision-making (MADM).
Methods: The basis of the procedure is to define the nodes of argument pairs and their parameters for the allowable multidimensional points. The parameters are implemented in the gradient method, the favorable directions method and the line search method. In the development of the procedure, the basics of the TOPSIS method for MADM with interval-given criteria weights were used, primarily due to the nonlinearity of the reference function.
Results: The paper elaborates the procedure of determining extreme and other admissible solutions of the reference function (boundary and basic solutions) and all vertices of the convex set of the function definition. This forms a complete graph of the function, i.e. the required solutions from the allowable set can be determined. A procedure for determining a set of solutions for defining a separating hyperplane of a set of function values has been developed; in this way, as a specific case, a set of solutions of partial stability of the variant is defined as MADM solutions. Adequate procedures have been proposed to eliminate the degeneration of the procedure (wedging and oscillation of the solution).
Conclusions: The most significant contribution of the paper is the definition of the nodes of argument pairs and their parameters which ensure the normalization condition in each node and for each allowable point, non-negativity of variables and independence of argument changes in nodes, within active constraints. An original procedure for determining function graphs has been developed. An appropriate real numerical example is given.
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