Multi-objective Optimization Proposal with Fuzzy Coefficients in both Constraints and Objective Functions.

Propuesta de optimización multiobjetivo con coeficientes difusos en restricciones y en funciones objetivo.AbstractFuzzy sets, and more specifically, fuzzy numbers can be a very suitable way to include uncertainty within the formulation and solution of linear problems with multiple goals. Goals in a decision problem do not need to be either maximized, or minimized, as in classical mathematical programming, but they are substituted by aspiration levels, and they need to be met in order to satisfy the decision-maker. Experience shows that it is easier for the decision-maker to formulate both objectives and constraints with fuzzy coefficients, rather than specify a defined quantity for the matrices A, b or g. This paper shows the versatility of a methodology that solves multi-objective linear problems, formulated with fuzzy coefficients. This conception becomes an alternative in contrast with the hard methodologies predominant in Operations Research, since the fuzzy approach allows the decision-maker to make uncertain assumptions both for the formulation and solution of optimization problems.Keywords: Fuzzy logic, multi-criteria analysis, triangular fuzzy numbers.ResumenLos conjuntos difusos y específicamente los números difusos constituyen una manera efectiva de incluir la incertidumbre en la formulación y solución de problemas lineales de optimización multiobjetivo. Las metas en un problema de decisión no necesitan ser maximizadas ni minimizadas, como ocurre en las herramientas clásicas de programación matemática, sino que se pueden sustituir por niveles de aspiración, las cuales constituyen las expectativas para un decisor. La experiencia demuestra que es más fácil para el decisor formular los objetivos y las restricciones en un problema con coeficientes difusos, en vez de simplemente especificar un número concreto en las matrices A, b ó g. Este artículo presenta la versatilidad de una formulación metodológica que permite resolver problemas multiobjetivo de tipo lineal, los cuales son formulados con coeficientes difusos. Esta concepción constituye una alternativa a las metodologías duras que dominan la investigación de operaciones, dado que la aproximación difusa permite que los decisores realicen presunciones inciertas en la formulación y solución en los problemas de optimización.Palabras Clave: Lógica difusa, análisis multiobjetivo, números triangulares difusos

Parametric formulation of the general integer linear programming problem

A parametric approach to the general integer programming problem is explored. If a solution to the general integer linear programming problem exists, it can be expressed as a convex combination of the extreme points of the convex polytope of the associated linear programming relaxation. The combination may or may not be unique for the convex polytope and will depend on the extreme points used in the determination. Therefore, a heuristic approach to solving the general integer programming problem can be taken by generating extreme points of the convex polytope and reformulating a mixed integer linear programming problem over these extreme points. This approach guarantees a feasible solution in a reasonable time frame. Further, such a technique can be used to provide quick lower bound information for an optimal search procedure