Relative Euclidean Distance With Application to TOPSIS and Estimation Performance Ranking

To extend existing distance metrics in the
$L^{p}$
space, we define a novel distance, named relative Euclidean distance (RED), and prove that it is positive definite, symmetric, and satisfies the triangle inequality. This distance has a value range of [0, 1]. We prove a uniqueness of it in the
$L^{p}$
space. The technique for order preference by similarity to ideal solution (TOPSIS) is widely used for multiple-attribute decision problems. The commonly used relative closeness measure in the TOPSIS is not a mathematical distance and thus does not have all the nice properties of such a distance. To improve the TOPSIS, particularly its relative closeness measure, we propose two measures, named optimistic distance (OD) and pessimistic distance (PD), based on our RED, and use them to measure the relative closeness in the TOPSIS. Both measures are positive definite, symmetric, and satisfy the triangle inequality. The best choice in OD (or PD) is on the Pareto frontier. Three examples of performance ranking are given to illustrate the usefulness of the TOPSIS with our proposed measures.