**2. Literature review**

In metric topology [2], the particular function of distance *D*(*a*,*b*) is used to assess the closeness of two points *a*, *b* as a real positive function that keeps three basic properties:


The general function of distance used to calculate the separation between two points is given as follows:

$$D(a,b) := \lim (\Sigma\_i (a\_i - b\_i)^n)^{1/n} (i = 1, \dots, n; n = \text{dimension of the space}).$$

*n k* à

When applying different values of *k*, different norms of distance appear:

a point of comparison (a threshold) that makes possible to compare or decide if our positions,

248 Applications and Theory of Analytic Hierarchy Process - Decision Making for Strategic Decisions

For our purposes, compatibility is defined as the proximity or closeness between vectors within

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

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

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

In metric topology [2], the particular function of distance *D*(*a*,*b*) is used to assess the closeness

The general function of distance used to calculate the separation between two points is given

( ) 1/ , lim( ( ) ) ( 1,..., ; dimension of the space). *n n D ab a b i nn* = S- = = *ii i*

between two different rules of measurement (two different points of view).

of two points *a*, *b* as a real positive function that keeps three basic properties:

**1.** *D*(*a*,*b*) > 0 and *D*(*a*,*b*) = 0 iff *a* = *b* (definition of zero distance)

**3.** *D*(*a*,*b*) + *D*(*b*,*c*) ≥ *D*(*a*,*c*) (triangular inequality)

system values, or priorities are really close.

vectors in cardinal measure bases (order topology) [1, 2].

really means close" in weighted environments.

**2. Literature review**

**2.** *D*(*a*,*b*) = *D*(*b*,*a*) (symmetry)

as follows:

a weighted space [1].

For *k* = 1, *D*(*a*,*b*) = Σ*<sup>i</sup>* abs(*ai* – *bi* ). Norm1, absolute norm, or path norm: this norm measures the distance from *a* to *b* within a 1D line, "walking" over the path, in one-line dimension.

For *k* = 2, *D*(*a*,*b*)= [(Σ*<sup>i</sup>* (ai – bi ) 2 )]1/2. Norm2 or Euclidean norm: this norm measures the distance from *a* to *b*, within a 2D plane (*X*– plane) getting the shortest path (the straight line).

For *k* = **+∞**, *D*(*a*,*b*) = Max*<sup>i</sup>* (abs(*ai* – *bi* )). Norm ∞ or Norm Max: this norm measures the distance from *a* to *b* within a ∞D hyperplane, getting the shortest path (the maximum coordinate) from all the possible paths.

In the field of statistics, we may note an interesting (and beautiful) case of distance calculation, which is known as distance of Mahalanobis [3], which meets the metric properties shown before. This distance takes into consideration parameters of statistics such as deviation and covariance (which can be assimilated to concepts of weight and dependence in the Analytic Hierarchy process/Analytic Network Process (AHP/ANP) world). Its formal presentation is:

1 1 (, ) ( ) ( ) with the matrix of covariance betweem , . *<sup>m</sup> d xy X Y X Y X Y* - - = -S - S

But, for a more simple case (without dependence), this formula can be written as:

$$d\_z(\mathbf{x}\_1, \mathbf{x}\_2) = \sqrt{(\frac{\text{x1}\,\text{l}\,\text{-}\,\text{x1}\,\text{2})}{\sigma\text{l}})^2 + (\frac{\text{x2}\,\text{l}\,\text{-}\,\text{x2}\,\text{2})}{\sigma\text{2}})^2} \text{or} \\ d\_z(X\_1, X\_2) = \sqrt{\left(X\_1 - X\_2\right)^\text{l}\,\text{S}^{-1}(X\_1 - X\_2)},$$

with *S*−1 the diagonal matrix with the standard deviation of variables *X*, *Y*.

It is interesting to see that the importance of the variable (to calculate distance) depends on the deviation value (bigger the deviation smaller the importance), that is, the importance of the variable does not depend on the variable itself, but just on the level of certainty on the variable (is this statement always true?).

We consider this approach into discussion, since factors such as weight and dependence are in the bases of AHP and ANP structures [4, 5]; but instead to understand and deal with probabilities and statistics (which by the way are not easy to build and later interpret), the idea here is to apply the natural way of thinking of human being which is based more on priorities than on probabilities. Indeed, we can manage the same information in a more comprehensive, complete, and easy to explain form by combining AHP/ANP with compatibility index *G*, and working with priorities, avoiding the needs of collecting big databases or understanding and interpreting complex statistic functions.

Thereby, the Multicriteria Decision Making (MCDM) approach through AHP/ANP method gives a very nice tool for our investigation and treatment of the knowledge and experience that experts possess in different fields, and at the same time staying within the decision-making domain (order topology domain), avoiding building huge, and costly databases where the knowledge about the individual behavior is lost.
