**3. Hypotheses/objectives**

In order topology measurement deals with dominance between preferences (intensity of preference), for instance *D*(*a,b*) = 3 means that dominance or intensity of preference of "*a*" over "*b*" is equal to 3, or that, *a* is three times more preferred than *b*. When talking about preferences, a *relative absolute ratio* scale is applied. *Relative*: because priority is a number created as a proportion of a total (percent or relative to the total) and has no need for an origin or predefined zero in the scale. *Absolute*: because it has no dimension since it is a relationship between two numbers of the same scale leaving the final number with no unit. *Ratio*: because it is built in a proportional type of scale (6 kg/3 kg = 2) [2].

So, making a general analogy between the two topologies, one might say that *"Metric Topology is to Distance as Order Topology is to Intensity"* [2, 6].

An equivalent concept of distance is presented to make a parallel between the three properties of distance of metric topology [1, 2].

This is applied in the order topology domain, considering a compatibility function *(*Eq. (1)) similar to distance function, but over vectors instead of real numbers.

Consideration: *A*, *B*, and *C* are priority vectors of positive coordinates and *Σiai* = *Σibi*= *Σici* = 1.

*G*(*A*,*B*) is the compatibility function expressed as:

$$G(A,B) = \% \sum \left( ai + bi \right) \frac{\text{Min}(ai, bi)}{\text{Max}(ai, bi)} \right) \tag{1}$$

This function presents [1, 2, 5, 6]:

**1.** 0 ≤ *G*(*A*,*B*) ≤ 1 (*nonnegative real number*)

The compatibility function, *G*, returns a nonnegative real number that lays in the 0–1 range. With *G*(*A*,*B*) = 0, if *A* and *B* are perpendicular vectors (*A*┴*B*), and represent the definition of total incompatibility between priority vectors *A* and *B* (*A*◦*B* = 0).

Also, *G*(*A*,*B*) = 1, if *A* and *B* are parallel vectors (*A* = *B* for normalized vectors), and represent the definition of total compatibility between priority vectors *A* and *B* (*A* ◦ *B* = 1).

**2.** *G*(*A*,*B*) = *G*(*B*,*A*) *(symmetry)*

Symmetry condition: the compatibility measured from *A* to *B* is equal to the compatibility measured from *B* to *A*.

Easy to proof, just interchanging *A* for *B* and *B* for *A* in the compatibility function *G*.

**3.** *G*(*A*,*B*) + *G*(*B*,*C*) ≥ *G*(*A*,*C*) (*triangular inequality*)

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

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

In order topology measurement deals with dominance between preferences (intensity of preference), for instance *D*(*a,b*) = 3 means that dominance or intensity of preference of "*a*" over "*b*" is equal to 3, or that, *a* is three times more preferred than *b*. When talking about preferences, a *relative absolute ratio* scale is applied. *Relative*: because priority is a number created as a proportion of a total (percent or relative to the total) and has no need for an origin or predefined zero in the scale. *Absolute*: because it has no dimension since it is a relationship between two numbers of the same scale leaving the final number with no unit. *Ratio*: because it is built in a

So, making a general analogy between the two topologies, one might say that *"Metric Topology*

An equivalent concept of distance is presented to make a parallel between the three properties

This is applied in the order topology domain, considering a compatibility function *(*Eq. (1))

Consideration: *A*, *B*, and *C* are priority vectors of positive coordinates and *Σiai* = *Σibi*= *Σici* = 1.

Max( , )

*ai*

*bi*

è ø <sup>å</sup> (1)

*ai bi ai bi*

The compatibility function, *G*, returns a nonnegative real number that lays in the 0–1 range. With *G*(*A*,*B*) = 0, if *A* and *B* are perpendicular vectors (*A*┴*B*), and represent the

Also, *G*(*A*,*B*) = 1, if *A* and *B* are parallel vectors (*A* = *B* for normalized vectors), and represent

(,) ½ ( ) Min( , )

definition of total incompatibility between priority vectors *A* and *B* (*A*◦*B* = 0).

the definition of total compatibility between priority vectors *A* and *B* (*A* ◦ *B* = 1).

*<sup>B</sup>* <sup>æ</sup> <sup>=</sup> <sup>ö</sup> ç ÷ <sup>+</sup>

similar to distance function, but over vectors instead of real numbers.

knowledge about the individual behavior is lost.

proportional type of scale (6 kg/3 kg = 2) [2].

of distance of metric topology [1, 2].

This function presents [1, 2, 5, 6]:

**2.** *G*(*A*,*B*) = *G*(*B*,*A*) *(symmetry)*

**1.** 0 ≤ *G*(*A*,*B*) ≤ 1 (*nonnegative real number*)

*is to Distance as Order Topology is to Intensity"* [2, 6].

*G*(*A*,*B*) is the compatibility function expressed as:

*G A*

**3. Hypotheses/objectives**

**4.** If *ACB* and *BCC* ⇔ *ACC* (*nontransitivity of compatibility*): if *A* is compatible with *B*, and *B* is compatible with *C*, does not imply that *A* is necessarily compatible with *C*.

For property 3, it is easy to prove that if *A*, *B*, and C are compatible priority vectors (i.e., 0.9 ≤ *Gi* ≤ 1.0 for *A*, *B*, and *C*), then property 3 is always satisfied. But, this property is also satisfied for the more relaxed (and interesting) condition where only two of the three vectors are compatible. For instance, if *A* is compatible with *B* (*G*(*A*,*B*) ≥ 0.9) and *A* is compatible with *C* (*G*(*A*,*C*) ≥ 0.9), or some other combination of *A*, *B*, and *C*, then condition 3 is also satisfied. This more relaxed condition allows compatible and noncompatible vectors to be combined while property 3 is still satisfied.

This situation can be geometrically viewed in the next figure.

**Figure 1** shows the compatibility neighborhood for *A*, in relation to *B* and *C*, with its minimum compatibility value of 0.9 represented by the radius of the circle (in the center, the compatibility reaches its maximum value of 1.0). Thus, *G*(*A*,*B*) = *G*(*A*,*C*) = 0.9 represents the minimum compatibility point, or the maximum distance for positions *B* and *C* to still be compatible with position *A*. Of course, *G*(*B*,*C*) < 0.9 that represents a noncompatible position for points *B* and *C*.

**Figure 1.** Maximum circle of compatibility for position A, related to B and C [2].

Notice that property 3, *G*(*A*,*B*) + *G*(*B*,*C*) ≥ *G*(*A*,*C*) is still valid, indeed any combination that one can be made will keep the inequality satisfied since if *C* gets closer to *A* (increasing the right side of the equation), then *G*(*B*,*C*) will also grow. The extreme case when *C* is over *A* (*G*(*A*,*C*) = 1.0), then *G*(*B*,*A*) + *G*(*B*,*C*) = 0.9 + 0.9 = 1.8 > 1.0 keeping the inequality satisfied [2].

We may also define incompatibility function as the arithmetic complement of the compatibil‐ ity:

Incompatibility = 1 – Compatibility.

Thus, incompatibility is equivalent to 1 – *G*. *By the way, the incompatibility concept is more close to the idea of distance, since the greater the distance the greater the incompatibility* [1, 2, 5, 6].

Two simple examples of this parallel between *D*(*x*,*y*) and *G*(*X*,*Y*) are given. But first, to make *D* and *G* functions comparable, absolute distance *D* must be transformed into relative terms as a percentage value since the priority vectors are normalized vectors for the *G* function. Thus, the maximum possible value for *D*<sup>1</sup> (Norm1) is 2 and for *D*<sup>2</sup> (Norm2) is 2, while performing the ratios with respect to the maximum possible value and obtaining *D* in relative terms as percentage of the maximum value.

For the first example, two different and very different vectors *A* and *B* with coordinate: {0.3, 0.7} with {0.7, 0.3} and {0.1, 0.9} with {0.9, 0.1} are considered.

Considering also incompatibility = 1 – Compatibility or 1 – *G*(*A*,*B*). Then:

Compatibility between *A* and *B* is shown by *G*(*A*,*B*) (*real positive value laying in 0– 1 range*).

Incompatibility between *A* and *B* is shown by 1 – *G*(*A*,*B*) (*real positive value laying in 0–1 range*).

**Table 1** shows the results of applying *D*<sup>1</sup> , *D*<sup>2</sup> , and (1 – *G*) functions.


**Table 1.** Evaluating distance and incompatibility for nonsimilar set of coordinates.

**Figure 2.** Two examples for distance and compatibility functions for far and very far A and B.

**Figure 2** shows in 2D Cartesian axes how far (incompatible) is *A* from *B* in both cases. Notice that for the case represented in brown (for *D* = 80% and *G* = 89%), vectors *A* and *B* are (geo‐ metrically) almost in a perpendicular position (*A* relative to *B*) [2].

$$\mathbf{l} - \mathbf{G} = \mathbf{S} \mathbf{\mathcal{T}} \mathbf{\tilde{\mathbf{\mathbf{\tilde{s}}}}} $$

Next, using the same procedure, we compare for similar and very similar *A* and *B* vectors with coordinates: {0.3, 0.7} compared with {0.4, 0.6} and {0.10, 0.90} compared with {0.11, 0.89}.

**Table 2** shows the results of applying *D*<sup>1</sup> , *D*<sup>2</sup> , and 1 – *G* (incompatibility = 1 – compatibility).


**Table 2.** Calculating distance and incompatibility for similar set of coordinates.

The trend of the results for *D* and G functions is the same in both cases, when increasing the distance or making vectors more perpendicular and when decreasing the distance or making the vectors more parallel. This is an interesting parallel to these concepts and their trends, considering that different concepts (distance and incompatibility in different ratio scales) are being used [2].
