**1. Introduction**

This chapter addresses the problem of measuring closeness in weighted environments (decisionmaking environments), using the concept of compatibility of priority vectors and value systems.

When using the concept of closeness, it comes in mind immediately what means to be close (when close really means close). Thus, when measuring closeness or proximity, we should add

a point of comparison (a threshold) that makes possible to compare or decide if our positions, system values, or priorities are really close.

For our purposes, compatibility is defined as the proximity or closeness between vectors within a weighted space [1].

We show a proposition for a compatibility index that can measure closeness in a weighted environment, thus can assess pattern recognition; medical diagnosis support measuring the degree of closeness between disease-diagnosis profiles, buyer-seller matching profiles; measuring the degree of closeness between house buyer and seller projects, or employment degree of matching; measuring the degree of closeness between a person's profile with the desired position profile; in curricula network design, conflict resolution; measuring closeness of two different value systems (the ways of thinking) by identifying and measuring the discrepancies, and in general measuring the degree of compatibility between any priority vectors in cardinal measure bases (order topology) [1, 2].

The chapter first presents some theory behind distance (measurement) and closeness concepts in different cases as well as a nice statistical view of distance. Then, it presents the concepts of scales, compatibility, compatibility index G, and some analogies among G and distance concept. Next, it shows a comparison with another compatibility indices present in the literature, reflecting the advantages of G in front of the others (especially within weighted environments). Then, it presents a necessary threshold, which allows establishing "when close really means close" in weighted environments.

Finally, three relatively simple examples are developed, each one presenting a different application for the compatibility index G. One for questioning if the order of choice should be a must to say if two rankings are compatible or not, one for quality testing (testing the Saaty's consistency index through compatibility index G), and one for measuring comparability between two different rules of measurement (two different points of view).
