Abstract

SANCHEZ-LARIOS, H  and  GUILLEN-BURGUETE, S.T.. Asymmetric and Non-Positive Definite Distance Functions Part II: Modeling. Ing. invest. y tecnol. [online]. 2009, vol.10, n.1, pp.75-83. ISSN 2594-0732.

Traditionally the distance functions involved in problems of Operations Research have been modeled using positive linear combinations of metrics Lp. Thus, the resulting distance functions are symmetric, uniforms and positive definite. Starting from a new definition of arc length, we propose a method formo deling generalized distance functions, that we call premetrics, which can be asymmetric, non uniform, and non positive definite. We show that every distance function satisfying the triangle inequality and having a continuous one-sided directional derivative can be modeled as a problem of calculus of variations. The "length" of a d-geodesic arc C(a,b) from a to b with respect to the premetric d (the d-length) can be negative, and therefore the d-distance from a to b may represent the minimum energy needed to move a mobile object from a to b. We illustrate our method with two examples.

Keywords : Distance functions; geodesics; variational calculus; facility location problem.