Asymmetric and Non-Positive Definite Distance Functions. Part II: Modeling

Abstract

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. STAR TING FROM A NEW DEFINITION OF ARC LENGTH, WE PROPOSE A METHOD FOR MODELING GENERALIZED DISTANCE FUNCTIONS, THAT WE CALL PRE METRICS, 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 THE REFORE 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.