**7. Three simple applications of compatibility index** *G*

### **7.1. Is the order of choice a must?**

In the second case (threshold over sorted figures), the ratio is much more closer (as expected to be), with a value of 1.16 (0.90 over 0.78), saying that order may help to improve compatibility but is not enough, it needs to consider the weights (not just the preference but the intensity of the preference) which is related to the values of the elements that belong to the vector, as well

Of course, this test should be carried out for a large number of experiments to have a more reliable response. A second test conducted for 225 experiments (15 people making 15 experi‐ ments each) has shown more or less the same initial results for average *G* value in both cases

Next, **Table 3** provides the meaning of ranges of compatibility in terms of index *G* and its

Compatibility at cardinal level of measurement (totally compatible)

as the angles of both vectors point to point (geometrically viewed as profiles).

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

**Description**

High 85–89.9 High compatibility (almost totally compatible)

Moderate 75–84.9 Moderate compatibility (compatibility at ordinal level)

(almost incompatible)

(totally incompatible)

Finally, another interesting way to illustrate the 90% as a good threshold for compatibility is the pattern recognition issue. Compatibility is the way to measure if a set of data (vector of priorities or profile of behavior) correspond to a recognized pattern or not. For instance, in the medical pattern recognition application, the diagnose profile (the pattern) is built with the intensity values of sign and symptoms that correctly describe the disease and then is compared with the sign and symptoms gathered from the patient, when these two profiles were about 90% or more of matching, then the physician was confident to say that the patient has the

When the matching between those profiles were 85–90%, the physicians in general agreed with the diagnoses offered by the software, but when the *G* values was below 85% (between 79– 84%), then the doctor sometime found trouble to recognize if the new sign and symptoms (the new patient's profile) was corresponding or not to the disease initially presented (nonconclu‐

with and without orders (± 0.78 and ± 0.50).

**Compatibility value range c(G%)** 

Very high ≥90% Very high compatibility

Low 65–74.9 Low level of compatibility Very low 55.1–64.9 Very low compatibility

Null (random) ≤55% Random compatibility

**Table 3.** Ranges of compatibilities and its meaning.

description.

**compatibility** 

**Degree cof**

disease.

We use to say that under the same decision problem, two compatible persons should make similar decisions. But, what do we mean when we say: "two compatible people should make similar decisions" [6, 7].

It means that they should make the same choice?

Take a look to the following case:

Two candidates: A and B for an election.

Three people, P1: choose candidate A, P2 and P3: choose candidate B.

P1 and P2 are moderate people, thus their intensity of preference for the candidates are for person 1: 55–45, for candidate A, and person 2: 45–55 for candidate B.

On the other hand, P3 is an extreme person, thus his intensity of preference is 5–95 for candidate B.

Is really P3 more compatible with P2 than P1 just because P3 makes the same choice of P2? (both have the same order of choice voting by candidate B). It seems that the order of choice is not the complete or final answer.

On the other hand, we know that in order topology, metric of decision means intensity of choice (dominance of A over B). So, compatibility is not related only to the simple order of choice, but also something more complex, more systemic, which is related with the intensity of choice.

Let us see the next numerical example (**Figure 10**).


**Figure 10.** Comparing intensities and order of choices of three people.

Suppose three people are having equal and different order of choices and its related priority vectors.

As we can see from **Figure 10**, the order of P1 is the same than the order of P3 and inverse of P2.

Order(P1) = Order(P3) ≠ Order(P2) (inverse order actually)

Considering just the above information, may we say that P1–P3 is closer than P1–P2?

Making the numbers (calculating G, for both combination P1–P3 and P1–P2), we found:

*G*(P1;P2) = 0.9 (≥90%), which implies that P1 and P2 have compatible choices (very high compatibility).

*G*(P1;P3)= 0.77 (<90%), which implies that P1 and P2 have noncompatible choices (moderate to low).

This is a very interesting result, considering that P2 have a totally inverted order of choice compared with P1. Yet, they are compatible people.

On the other side, P1 and P3, which have the same order of choices, are not compatible people.

Hence, it is very important to be able to measure the degree of compatibility (alignment) in a reliable way.

Note: in Alice in wonderland of Charles Lutwidge Dodgson (Lewis Carroll) is a phrase saying: "I tell you, sometimes 1-2-3 might look more like 3-2-1 than 1-2-3" [6, 7].

#### **7.2. Mixing consistency and compatibility indices in a metric quality test drive**

A different and interesting application of *G* is possible when it is used to check the quality of a metric.

When it is possible to compare a metric obtained with some method with the expected or actual metric, then the compatibility index *G* represents a great tool to test and verify the quality of the created metric.

Suppose (for instance), we want to measure the quality of the following simple example.

#### *7.2.1. The problem (the criticism)*

Setting an hypothetic problem (a criticism made by some critic person) about the quality of the consistency index in pair comparison matrices (Saaty's index) [4, 5].

The hypothetical critic says: the index of consistency (Saaty's index) is wrong, since it lest pass through values (comparisons) that are not acceptable by common sense.

Suppose, three equal bars of same length like the ones in **Figure 11**.

**Figure 11.** Bar comparisons.

**Figure 10.** Comparing intensities and order of choices of three people.

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

Order(P1) = Order(P3) ≠ Order(P2) (inverse order actually)

compared with P1. Yet, they are compatible people.

vectors.

compatibility).

reliable way.

a metric.

the created metric.

to low).

P2.

Suppose three people are having equal and different order of choices and its related priority

As we can see from **Figure 10**, the order of P1 is the same than the order of P3 and inverse of

Considering just the above information, may we say that P1–P3 is closer than P1–P2?

Making the numbers (calculating G, for both combination P1–P3 and P1–P2), we found:

*G*(P1;P2) = 0.9 (≥90%), which implies that P1 and P2 have compatible choices (very high

*G*(P1;P3)= 0.77 (<90%), which implies that P1 and P2 have noncompatible choices (moderate

This is a very interesting result, considering that P2 have a totally inverted order of choice

On the other side, P1 and P3, which have the same order of choices, are not compatible people.

Hence, it is very important to be able to measure the degree of compatibility (alignment) in a

Note: in Alice in wonderland of Charles Lutwidge Dodgson (Lewis Carroll) is a phrase saying:

A different and interesting application of *G* is possible when it is used to check the quality of

When it is possible to compare a metric obtained with some method with the expected or actual metric, then the compatibility index *G* represents a great tool to test and verify the quality of

Suppose (for instance), we want to measure the quality of the following simple example.

"I tell you, sometimes 1-2-3 might look more like 3-2-1 than 1-2-3" [6, 7].

**7.2. Mixing consistency and compatibility indices in a metric quality test drive**


**Figure 12.** Possible application of index *G*. (a) Bar comparisons. (b) Bar comparisons.

Of course, the correct matrix comparison for this situation is the following (consistent) comparison matrix:

The obvious (correct) priority vector "*w*" is: 1/3, 1/3, 1/3, with 100% of consistency (CR = 0).

Suppose now that (due to some visualization mistake), the new appreciation about the bars is:

The new (perturbed) priority vector *w*\* is 0.4126, 0.3275, 0.2599, with CR = 0.05 (95% of consistency), which according to the theory is the maximum acceptable CR for a 3 × 3 com‐ parison matrix.

The critic claims that the *A*–*C* bars comparison has a 100% of difference (100% of error), which is not an acceptable/tolerable error (easy to see even at naked eye).

Also, the global error in the priority vectors is 15.85%, calculated with the common formula: *e* = abs(*u* – *v*)/*v*, for each coordinate and then take the average of the coordinates.

But, Saaty's consistency index says that (CR = 95%), which is tolerable for a 3 × 3 comparison matrix. Hence, the critic claims that Saaty's consistency index is wrong.

#### *7.2.2. The response*

The already described problem has at least two big misunderstanding issues:

First:

**•** The CR (the Saaty's index of consistency) comes from the eigenvalue-eigenvector problem, so it is a systemic approach (does not care in a particular comparison) [4, 5].

Second:

**•** The possible error should be measured by its final result (the resulting metric), not in the prior or middle steps.

#### *7.2.3. The explanation*

The first misunderstanding is explained by itself.

For the second one, before any calculation, we need to understand what kind of numbers are we dealing with (in what environment we are working), because it is not the same to be close to a big priority than to a small one. This is a weighted environment and the measure of the closeness (proximity) has to consider this situation.

We must work in the order topology domain to correctly measure the proximity (closeness) on this environment. To do this correctly, two aspects of the information must to be considered: the intensity (the weight or priority) and the degree of deviation between the priority vectors (the projection between the vectors). The index that takes good care of these two factors simultaneously is the compatibility index *G*.

Summarizing, the vectors of correct and perturbed metric are:


The basic question here is how close is the approximated metric to the correct metric?

Evaluating *G*(correct – perturbed), the *G* value obtained is 85.72%, which in numerical terms represents almost compatible metrics (high compatibility).

As explained at the end of point 5, *G* = 90% is a threshold to consider two priority vectors as compatible vectors. Also, *G* = 85% is an acceptable lower limit value (high compatibility).

Hence, the two metrics are relatively close (close enough considering that they are not physical measures).

Of course, better consistency can be achieved.

The question is do we really obtain a better result when being totally consistent?

And the answer is: probably no. Because, in real problems we never have the "real" answer (the true metric to contrast). Experience shows that pursuing consistent metrics per se may provide less sustained results.

For instance, in the presented problem one could answer that *A* – *B* = 2, *A* – *C* = 2, and *B* – C = 1, and he/she would be totally consistent, but consistently wrong.

By the way, the new priority vector would be *w*\*\* = (0.5, 0.25, 0.25), with CR = 0 (totally consistent), and *G* = 71.5%, which means incompatible vectors (low compatibility). Thus, a totally consistent metric is incompatible with the correct result.

So, at the end it is better to be approximately correct than consistently wrong.

(The consistency index is just a thermometer not a goal).

But, Saaty's consistency index says that (CR = 95%), which is tolerable for a 3 × 3 comparison

**•** The CR (the Saaty's index of consistency) comes from the eigenvalue-eigenvector problem,

**•** The possible error should be measured by its final result (the resulting metric), not in the

For the second one, before any calculation, we need to understand what kind of numbers are we dealing with (in what environment we are working), because it is not the same to be close to a big priority than to a small one. This is a weighted environment and the measure of the

We must work in the order topology domain to correctly measure the proximity (closeness) on this environment. To do this correctly, two aspects of the information must to be considered: the intensity (the weight or priority) and the degree of deviation between the priority vectors (the projection between the vectors). The index that takes good care of these two factors

Correct metric (priority vector): 0.3333 0.3333 0.3333 Perturbed or approximated metric (priority vector): 0.4126 0.3275 0.2599

The basic question here is how close is the approximated metric to the correct metric?

Evaluating *G*(correct – perturbed), the *G* value obtained is 85.72%, which in numerical terms

As explained at the end of point 5, *G* = 90% is a threshold to consider two priority vectors as compatible vectors. Also, *G* = 85% is an acceptable lower limit value (high compatibility).

Hence, the two metrics are relatively close (close enough considering that they are not physical

matrix. Hence, the critic claims that Saaty's consistency index is wrong.

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

The already described problem has at least two big misunderstanding issues:

so it is a systemic approach (does not care in a particular comparison) [4, 5].

*7.2.2. The response*

prior or middle steps.

The first misunderstanding is explained by itself.

closeness (proximity) has to consider this situation.

Summarizing, the vectors of correct and perturbed metric are:

represents almost compatible metrics (high compatibility).

Of course, better consistency can be achieved.

simultaneously is the compatibility index *G*.

*7.2.3. The explanation*

First:

Second:

measures).

*Note 1: CR == 0.05 is the maximum acceptable value for inconsistency thus, I cannot go further with the bar comparison number (that means I cannot put a 3 instead of 2 in the cell (1, 3)).*

*Note 2: If metric B is compatible with metric A, then it is possible to use metric B as a good approximation of A. This is a useful property when metric A is not available (the most of the times we do not know the correct/exact metric).*

*Note 3: The same exercise was performed from 4 × 4 to 9 × 9 matrices that is, putting a value (n – 1) in the cell-position: (1, n), (n = matrix dimension), obtaining similar results (some times even better).*

#### **7.3. Using compatibility index** *G* **to measure the comparability (closeness) of two different metrics**

When you have two rules of measurement in the same structure of decision *(two different system values)*, how to combine and or compare the outcome of both?


#### Third: Measuring the compatibility *(the topological or closeness option)*

The first two options are long known and described in the literature [1, 2, 4, 5]. The third one is based on the compatibility principle and presents the following five advantages [6]:


**5.** Use the numerical option (geometric mean) when and where it is necessary (modifying values in the places where is more necessary in terms of efficiency).

For a better explanation, we will use an example to illustrate in details the already exposed idea.

Suppose that we are in front the following problem, a mine company needs to change its shiftwork system, from the actual 6 × 1 × 2 × 3 (from family of shifts of eight working hours), to the new (desired) shift-work system 4 × 4 (from family of shifts of 12 working hours).

6 × 1 × 2 × 3 means a shift of 6 days of work 1 day off, 6 days of work 2 out and 6 days of work 3 out, with 8 h per day, every week). After calculation, we have 144 of total working hours in 24 days of each working cycle.

4 × 4 means 4 days of work followed by 4 days off with 12 h per day every week). After calculation, we have again 144 of total working hours in 24 days of each working cycle.

Are those shifts equivalents? How to know what shift work is better? (or less risky, since any shift is bad by essence).

Even if the total labor hours is the same (in terms of quantity of working hours), the shift-works are not equivalent (in terms of quality of life and production), it depends on a bunch of interdependent variables (numbers of working hours per day, the entry time, the number of free days per year, the number of complete weekends per year, the number of nights per shift, the number of changes day/night/day per cycle, the number of sleeping hours, the opportunity of sleep, among many others), it depends also on how those variables are settled down and, of course, the weight (the importance that each variable has), which in time depends on what people you ask for (workers, managers, stakeholders, owners, family, or even the people that live surrounding the mine).

**Figure 13.** Comparing two rules of measurement.

Suppose now we have two evaluation scenarios:

First: The decision rule (DR) of measurement is built with the people that work in the 6 × 1 × 2 × 3 shift-work (weighting the variables involved in the rule of measurement of this shift), since they know better how it work their shift-work.

Second: The DR of measurement is built with the people that work in the 4 × 4 build weighting the variables involved in rule of measurement of this shift, since they know better their shiftwork.

As the process concludes, we end with two different outputs: for the 6 × 1 × 2 × 3 shift (using the first rule of measurement), we have an impact index of 0.33 (33%), while for the second shift (with the second rule), we have an impact index of: 0.37 (37%). Of course, I cannot say that shift 6 × 1 × 2 × 3 is better than 4 × 4 just because it has a lower impact (0.33 < 0.37), since they were built with different rules of measurement.

At the end, we have two DRs for the same problem; of course the question is not what rule is better, but how to make comparable both DRs.

One option could be to agree to use the same DR for both cases (the consensus, or verbal solution). But, this is not an option since the knowledge is located in different groups of people and is specific for each case; also they did not feel comfortable making this agreement.

Another option could be to take the geometric average (GM) of both rules and work with it as the final rule. Even if this could be a possibility, we really do not know what we are doing when combining or mixing both rules (as an analogy we cannot just combine one rule of measurement in meters with one rule in inches). We need to know first if both DRs are comparable.

Graphically:

**5.** Use the numerical option (geometric mean) when and where it is necessary (modifying

For a better explanation, we will use an example to illustrate in details the already exposed

Suppose that we are in front the following problem, a mine company needs to change its shiftwork system, from the actual 6 × 1 × 2 × 3 (from family of shifts of eight working hours), to the

6 × 1 × 2 × 3 means a shift of 6 days of work 1 day off, 6 days of work 2 out and 6 days of work 3 out, with 8 h per day, every week). After calculation, we have 144 of total working hours in

4 × 4 means 4 days of work followed by 4 days off with 12 h per day every week). After calculation, we have again 144 of total working hours in 24 days of each working cycle.

Are those shifts equivalents? How to know what shift work is better? (or less risky, since any

Even if the total labor hours is the same (in terms of quantity of working hours), the shift-works are not equivalent (in terms of quality of life and production), it depends on a bunch of interdependent variables (numbers of working hours per day, the entry time, the number of free days per year, the number of complete weekends per year, the number of nights per shift, the number of changes day/night/day per cycle, the number of sleeping hours, the opportunity of sleep, among many others), it depends also on how those variables are settled down and, of course, the weight (the importance that each variable has), which in time depends on what people you ask for (workers, managers, stakeholders, owners, family, or even the people that

values in the places where is more necessary in terms of efficiency).

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

new (desired) shift-work system 4 × 4 (from family of shifts of 12 working hours).

idea.

24 days of each working cycle.

shift is bad by essence).

live surrounding the mine).

**Figure 13.** Comparing two rules of measurement.

The problem statement can be placed as "are these two set of decision criteria (profiles gray and black of **Figure 13**) comparable/compatible among them?

**Figure 14.** (a) Compatibility as a concept of distance. (b) Compatibility as a concept of distance. (c) Compatibility as a concept of distance.

Recall that they represent two different DRs. Rule one (the black bars) represents the rule of measurement for 6 × 1 × 2 × 3 shift-work and is formed by four criteria (extracted from the global rule). Rule two (in gray) represents the rule of measurement for 4 × 4 shift-work and is formed by the same four criteria, but with different intensities.

By the way, when saying comparable DR, it means compatible DR (equivalents of measure), that is we can measure the level of risk of the alternatives with any of the two rules described above.

To establish if both DR are equivalent, it is necessary to calculate *G* of both profiles:

*G*(Profile gray – Profile black) = 0.91 (91%) ≥ 90%, which means they are compatible (sometimes, we can take 85% as an acceptable limit point of compatibility, but no less).

Thus, we can use any of the two DRs to measure the effect of the changing shift-work. Moreover, we may use the geometric mean (GM) of both DRs as the final rule, but now knowing that we are combining rules that are compatible in fact, we are not mixing two far away points of view.

This is a relevant issue, which helps a lot when working with different groups of decision making.

But, how to make both DR comparable when they are not compatible? (or what rule to use to measure the alternatives?)

There are four different cases of compatibility for DR where "*G*" can be applied:

**Case 1:** Case 1 is the case already described where the DR of each group decision making is compatible.

In this case, it is possible to use any of the two rules or (still better) use the geometric mean of both DRs.

Graphically, it can be seen as:

Analytically: *G*(DR1,GM) and *G*(DR2,GM) > *G*(DR1,DR2) > 0.90 (GM rule is better than any of the other two).

When measuring the alternatives with this final DR, make the results to be comparable.

**Case 2:** The DRs of both persons (or groups) are not comparable (compatible) among them, but are compatible with the GM:

In this case, take GM of both rules then measure the compatibility of each DR with regard to the GM rule. If both initial DRs are compatible with the GM rule, then you may use the GM rule as the final rule.

Graphically:

**Figure 14.** (a) Compatibility as a concept of distance. (b) Compatibility as a concept of distance. (c) Compatibility as a

Recall that they represent two different DRs. Rule one (the black bars) represents the rule of measurement for 6 × 1 × 2 × 3 shift-work and is formed by four criteria (extracted from the global rule). Rule two (in gray) represents the rule of measurement for 4 × 4 shift-work and is

By the way, when saying comparable DR, it means compatible DR (equivalents of measure), that is we can measure the level of risk of the alternatives with any of the two rules described

*G*(Profile gray – Profile black) = 0.91 (91%) ≥ 90%, which means they are compatible (sometimes,

Thus, we can use any of the two DRs to measure the effect of the changing shift-work. Moreover, we may use the geometric mean (GM) of both DRs as the final rule, but now knowing that we are combining rules that are compatible in fact, we are not mixing two far

This is a relevant issue, which helps a lot when working with different groups of decision

But, how to make both DR comparable when they are not compatible? (or what rule to use to

**Case 1:** Case 1 is the case already described where the DR of each group decision making is

In this case, it is possible to use any of the two rules or (still better) use the geometric mean of

Analytically: *G*(DR1,GM) and *G*(DR2,GM) > *G*(DR1,DR2) > 0.90 (GM rule is better than any of

When measuring the alternatives with this final DR, make the results to be comparable.

There are four different cases of compatibility for DR where "*G*" can be applied:

To establish if both DR are equivalent, it is necessary to calculate *G* of both profiles:

we can take 85% as an acceptable limit point of compatibility, but no less).

formed by the same four criteria, but with different intensities.

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

concept of distance.

above.

making.

compatible.

both DRs.

the other two).

away points of view.

measure the alternatives?)

Graphically, it can be seen as:

Analytically: *G*(DR1,GM) and *G*(DR2,GM) ≥ 0.9 > *G*(DR1,DR2)

When measuring the alternatives with this final DR, make the results to be comparable.

**Case 3:** The DRs of both persons are not compatible among them, but one is compatible with the GM.

In this case, look at the compatibility profile of GM with P2, G(GM;P2), for the position with the smallest number and proceed in the next sequence.

First, check if it is any "entry" error in the calculation process of P2 profile (some large inconsistency, or inverse entry in the comparison matrix).

Second, check if the comparisons in the matrix associated to that position is what P2 really meant to say.

Third, suggesting P2 to test acceptable numbers that produce a bigger *G*(getting closer to GM rule), until you can fall in Case 2.

Graphically:

Analytically: *G*(GM,P1) > 0.9 > *G*(P1,P2); *G*(GM,P2) > *G*(P1,P2) < 0.9

When measuring the alternatives with this final DR, make the results to be comparable.

There is a fourth case. The case where the initial DRs are not compatible among them and any of two DRs are also not compatible with GM rule. This is the toughest case, since P1 and P2 have very different points of view.

The suggestion for this case is as follows: Revise the structure of the model to find some lost criterion or border condition. The weights of the criteria have to be checked, and support information of P1 and P2 opinions revised.

If there are no changes with the initial position, you may (as last resource) apply (or impose) the GM as the final rule, but probably both people (or group) may feel not fully represented by that imposed DR.
