**5. Comparing G with other compatibility indices in the literature**

It should be noted that when using other compatibility index formulae, for instance, the classic dot product and dividing the result by "n" (*"n" is* the vector dimension), this operation is like taking the average instead of weighting to evaluate the incompatibility, the result of this operation applied on the last example of vectors *A* and *B* is: (0.1/0.9 + 0.9/0.1)/2 = 4.55 (or 355% of incompatibility or deviation from *n*) [1, 2, 6, 7].

Doing the same in a matrix environment, that is forming the Hadamard product [4, 5] and dividing by *n*<sup>2</sup> instead of "*n*" (again taking double average instead of weighting) to assess the deviation, the result is 83.012/22 = 20.75 (20.75 – 1 = 19.75 or 1975% of incompatibility or deviation from *n*<sup>2</sup> , presenting both situations a singularity problem (not bounded function). So, they do not seem to be adequate compatibility indices.

Another formula to evaluate incompatibility is the David Hilbert's formula [8, 9], whose expression is log(Max*<sup>i</sup>* (*a*/*b*)/Min*<sup>i</sup>* (*a*/*b*)).

When applying this expression, the formula returns the value of 1.908 (191% of incompatibil‐ ity). Thus, it also presents a singularity problem (results above the 100% value tough to be interpreted).

Note: *In this research, it was not included compatibility indices based on ordinal scale, because the ordinal scale is not able to respond adequately to the more rich and complex concept of proximity (distance). In fact, the order topology shows clearly that is the intensity (not the ranking), what really matter to establish proximity between vectors.*


A simple example is shown in **Figure 5** [10].

**Figure 5.** Example of relative electric consumption of household appliances.

#### **5.1. Comparing performance of** *G* **with other compatibility indices**

Suppose the next situation, having an AHP and actual priority vectors for the relative electric consumption of household appliances.

The AHP priority vector is the priority vector that is obtained from the 7 × 7 pair comparison matrix of the alternatives, and actual priority vector is using the direct relation between the consumption of each alternative related to the total.

We have:

The same for coordinate 2:

dividing by *n*<sup>2</sup>

interpreted).

deviation from *n*<sup>2</sup>

expression is log(Max*<sup>i</sup>*

deviation, the result is 83.012/22

of incompatibility or deviation from *n*) [1, 2, 6, 7].

So, they do not seem to be adequate compatibility indices.

(*a*/*b*)).

(*a*/*b*)/Min*<sup>i</sup>*

*matter to establish proximity between vectors.*

A simple example is shown in **Figure 5** [10].

**Figure 5.** Example of relative electric consumption of household appliances.

(Min/Max)2 = 0.111, and *G*(*A*,*B*) = ½((0.1 + 0.9)0.111 + (0.9 + 0.1)0.111) = 0.111 (11.1% of com‐ patibility or 88.9% ≫ 10% of incompatibility). This is also an expected outcome since geomet‐ rically they are almost perpendicular vectors (vector *A* has almost no projection over *B*).

It should be noted that when using other compatibility index formulae, for instance, the classic dot product and dividing the result by "n" (*"n" is* the vector dimension), this operation is like taking the average instead of weighting to evaluate the incompatibility, the result of this operation applied on the last example of vectors *A* and *B* is: (0.1/0.9 + 0.9/0.1)/2 = 4.55 (or 355%

Doing the same in a matrix environment, that is forming the Hadamard product [4, 5] and

Another formula to evaluate incompatibility is the David Hilbert's formula [8, 9], whose

When applying this expression, the formula returns the value of 1.908 (191% of incompatibil‐ ity). Thus, it also presents a singularity problem (results above the 100% value tough to be

Note: *In this research, it was not included compatibility indices based on ordinal scale, because the ordinal scale is not able to respond adequately to the more rich and complex concept of proximity (distance). In fact, the order topology shows clearly that is the intensity (not the ranking), what really*

instead of "*n*" (again taking double average instead of weighting) to assess the

, presenting both situations a singularity problem (not bounded function).

= 20.75 (20.75 – 1 = 19.75 or 1975% of incompatibility or

**5. Comparing G with other compatibility indices in the literature**

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


So, AHP and actual vectors are noncompatible under *S*, *H*, or IVP indices, yet, compatible under *G* index. However, AHP and actual values are compatible vectors indeed (as expected), and *G* is the only index that captures correctly this condition of closeness between vectors [1, 2, 5, 6].

The first three formulae present a divergence process (singularities). As vector *A* deviates from vector *B* (being more and more incompatible), they become closer to perpendicular vectors.

The *G* function becomes the only formula capable of correctly assessing the compatibility without falling in a singularity (*divergence*), or remaining immobilized, as a standard distance calculation does when the absolute difference between the coordinates is kept constant. In fact, a complete study1 of the behavior of different compatibility indices is presented in **Table 4**<sup>2</sup> [2, 6, 7].

The study was made initially for a 2D space (vectors of two coordinates), since the idea was to perform a sensitive analysis and observe the patterns of behavior of different compatibility indices in two special situations:


<sup>1</sup> Claudio Garuti, "When close really means close?" Paper of Compatibility. Presented in the ISAHP9 Symposium and Proceeding. Viña del Mar-Chile, 2007.

<sup>2</sup> The different compatibility indexes analyzed were taken from the literature and from linear algebra definitions.


**Table 4.** Possible field of applications for index *G*.

**Figure 6** summarizes the compatibility indices that exist, adding the option of Euclidian norm (the classic distance calculation based on Norm2), normalized by its maximum possible value of to present the results in a percentage format.

**Figure 6.** Definition of different formulas for compatibility assessment.

Six formulae in a 2D vector for seven cases for two different trends (parallel trend and perpendicular trend) were tested. The results are shown in **Figure 7**.
