**8. Conclusions**

There are two global conclusions on this study:

First, we need some kind of index for distance/alignment measurement in weighted environ‐ ments (order topology domain), in order to mathematically define if two profiles (or behaviors) are really close.

This index (the compatibility index) makes possible:


Second, this analysis shows that the only compatibility index that performs correctly for every case is the *G* index, keeping the outcome always in the 0–100% range (like Euclidean formula), and this is an important condition, since any value out of the 0–100% range would be difficult to interpret (and the beginning of a possible divergence).

It is also important to note the *G* and Euclidean outcomes in **Table 4** are close, but *G* is much more accurate or sensitive to changes, because *G* is not based on absolute differences (Δ*xi* ) as distance does, but on relative absolute ratio scales. We have to remember that we are working on ratio scales (absolute ratio scale to be precise). In fact, the Euclidean distance calculation shows no differences in the distance of parallel trend from case 1 to 6 (see **Figure 8**), Euclideanbased index cannot detect the difference in the compatibility value among those cases because the absolute difference of the coordinates remains the same.

Therefore, with the Euclidian-based index one may reach the wrong conclusion that no difference exists for vector compatibility from cases 1 to 6 in the figure (the first case study is as incompatible as 2, 3, 4, 5, or 6), which is not an expected result. This unexpected behavior occurs because the Euclidean norm is based on differences, and also because it is not concerned about the weights of the coordinates and the projection between the priority vectors. It is important to remember that the numbers inside the priority vector represent preferences. Hence, in terms of proximity, it is better to be close to a big preference (big coordinate) than to the small one and this issue is better resolved in ratio scales.

Other tests made in greater spaces (3D to 10D) show the same trend. Even more, the bigger the space dimension the greater the likelihood of finding singularity points for the others formulae, in both trends parallel and perpendicular.

It is interesting to note that function *G* is not the simple dot product, since it depends on two different dimensional factors. On the one hand, it is the intensity of preference (related with the weight of the element), and on the other hand it is the angle of projection between the vectors (the profiles).

It means that *G* is a function of the intensity of preference (*I*) and the angle of projection (*α* between the priority vectors, that is *G*= *f*(*I*,*α*).

Clearly, the *G* function is not the simple dot product (as normally defined), but something more complex and rich of information.

It is also important to note that both data (intensity and angle) are normally implicit in the coordinates of the priority vectors and have to be correctly extracted; it depends also if the priorities are presented in relative measurement (RM) or absolute measurement (AM) format.

The possible applications of this index are huge and for different fields, as an example, for social and management sciences the possibilities are.

#### **8.1. Concluding remarks**

First, we need some kind of index for distance/alignment measurement in weighted environ‐ ments (order topology domain), in order to mathematically define if two profiles (or behaviors)

**•** have one more tool for conflict resolution on group decision making to achieve a possible

**•** membership analysis (closeness analysis to estimate if an element belong to one set of

Second, this analysis shows that the only compatibility index that performs correctly for every case is the *G* index, keeping the outcome always in the 0–100% range (like Euclidean formula), and this is an important condition, since any value out of the 0–100% range would be difficult

It is also important to note the *G* and Euclidean outcomes in **Table 4** are close, but *G* is much more accurate or sensitive to changes, because *G* is not based on absolute differences (Δ*xi*

distance does, but on relative absolute ratio scales. We have to remember that we are working on ratio scales (absolute ratio scale to be precise). In fact, the Euclidean distance calculation shows no differences in the distance of parallel trend from case 1 to 6 (see **Figure 8**), Euclideanbased index cannot detect the difference in the compatibility value among those cases because

Therefore, with the Euclidian-based index one may reach the wrong conclusion that no difference exists for vector compatibility from cases 1 to 6 in the figure (the first case study is as incompatible as 2, 3, 4, 5, or 6), which is not an expected result. This unexpected behavior occurs because the Euclidean norm is based on differences, and also because it is not concerned about the weights of the coordinates and the projection between the priority vectors. It is important to remember that the numbers inside the priority vector represent preferences. Hence, in terms of proximity, it is better to be close to a big preference (big coordinate) than

Other tests made in greater spaces (3D to 10D) show the same trend. Even more, the bigger the space dimension the greater the likelihood of finding singularity points for the others

It is interesting to note that function *G* is not the simple dot product, since it depends on two different dimensional factors. On the one hand, it is the intensity of preference (related with the weight of the element), and on the other hand it is the angle of projection between the

agreement considering that those profiles may represent system values **•** a pattern recognition process (assessing how close is one pattern to another)

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

are really close.

**•** a matching analysis process

**•** making a better benchmarking

elements or another).

This index (the compatibility index) makes possible:

to interpret (and the beginning of a possible divergence).

the absolute difference of the coordinates remains the same.

to the small one and this issue is better resolved in ratio scales.

formulae, in both trends parallel and perpendicular.

vectors (the profiles).

**•** analysis and test of quality of the results

There are many different applications for index *G*, next a summary with all possibilities that *G* may have:

#### **• Compatibility of systems value:**

*G* is an index able to be used in social and management sciences to measure compatibility of group decision-making (DMs) intra- and intergroups. The expression of *G* for this case is *G*(DM1 – DM2), which means level of compatibility (closeness) between DM1 and DM2. With DM1 and DM2, the decision's metric of each decision maker.

#### **• Compatibility for quality test:**

*G* can help to assess the quality of a built decision metric. As presented in point 7.2, *G* may help to evaluate the quality of any new metric based in a ratio scale. The result is achieved comparing the new metric with some standard or with an already known result.

#### **• Profiles alignment**

) as

*G* can help to establish if two different profiles are aligned. In general, it is not an easy task to know if two complex profiles are aligned, especially when the profiles are complex with many variables, with different importance and different behavior on each one. This is the case when try to measure the degree of matching between a medical diagnose and a list of diseases, or the degree of matching between a sale project and its possible buyers and many other similar cases.

#### *Formula expression: G(Profile1 – Profile2)* **=**

### **• Compatibility for comparability**

*G* can help to establish if two different measures are or not comparable, one relevant point when compare numbers from different outcomes is to know if those numbers are compa‐ rable or not. For instance, if I know that the impact of strategy *A* is 0.3 and the impact of strategy *B* is 0.6, I cannot say that strategy *A* impacts twice than strategy *B*, at least both strategies were measured with exactly the same rule. But, for many reasons, sometimes that is not possible. In that case I need to know if the rules of measurements are compatible among them. If so, it is possible to compare both numbers, else it is not possible.

#### **• Compatibility for sensitive analysis and threshold**

*G* can help to establish the degree of membership or the trend for membership (tendency) of an alternative. The idea is equivalent to the classic sensitive analysis when making small changes in the variables. The change resulting in the *G* value (before and after the sensitive analysis) would show where the alternative is more likely to belong (trend of belonging).
