**4. Research design/methodology**

One way to calculate compatibility in a general form is by using the inner or scalar vector product, defined as *A*◦*B* = |*A*||*B*|cos *α*. This expression of dot product is preferable to the Cartesian version, since it highlights the relevance of the projection concept represented by cos *α* and also because when working with normalized vectors the expression *A*◦*B* becomes equal to cos *α*, which shows that the projection part of the dot product is the most relevant part.

### **4.1. Definitions and conditions [2]**

Assuming:

the maximum possible value for *D*<sup>1</sup> (Norm1) is 2 and for *D*<sup>2</sup>

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

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:

percentage of the maximum value.

**Table 1** shows the results of applying *D*<sup>1</sup>

**Distance** *A***–***B* **in Norm1 (normalized)** 

*1 range*).

*D***<sup>1</sup> (***a***.***b***)**

= Sum (Abs(differences)/2) D2

*range*).

*A***,** *B*

**Coordinates** 

*A* = {0.3, 0.7} *B* = {0.7, 0.3}

*A* = {0.1, 0.9} *B* = {0.9, 0.1}

D1

the ratios with respect to the maximum possible value and obtaining *D* in relative terms as

For the first example, two different and very different vectors *A* and *B* with coordinate: {0.3,

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

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

**Distance** *A***–***B* **in Norm2 (normalized)** 

0.8/2 = 0.4 (40%) 0.4**√**2/**√**2 = 40% 1 – (0.3/0.7) = 57%

1.6/2 = 0.8 (80%) 0.8**√**2/**√**2 = 80% 1 – (0.1/0.9) = 89%

, and (1 – *G*) functions.

, *D*<sup>2</sup>

*D***<sup>2</sup> (***a***,***b***)**

= Sqroot(Sum(Square(differences))2)

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

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

(Norm2) is 2, while performing

**Incompatibility = 1 – G(***A***,***B***)** 

**A.** Two normalized vectors are closer (compatible), when the angle (*α*) formed by both vectors on the hyperplane is about 0° or cos *α* is near to 1. From a geometric point of view, they will be represented by parallel or nearly parallel vectors. In this case, they will be defined as compatible vectors.

**B.** Two normalized vectors are not closer (not compatible) when the angle (*α*) formed by both vectors on the hyperplane is near 90° or cos *α* is near to 0. From a geometric point of view, they will be represented by perpendicular or nearly perpendicular vectors. In this case, they will be defined as noncompatible vectors.

**Figure 3** [1, 2] shows the geometric interpretation of vector compatibility. Therefore, there is an operative way to measure compatibility in terms of vector projection. This interpretation of dot product will be very useful for the purposes of finding a compatibility measurement of two priority vectors in the domain of order topology.

**Figure 3.** Geometric interpretation of vector compatibility in terms of its projection.

Since the space is weighted (we are working in weighted environment), it is also necessary to weight each projection (each cos *α<sup>i</sup>* ) and to take into account the changes of the angle and the weight coordinate by coordinate (coordinate "*i*" may have a different projection and weight than coordinate "*i* + 1"). Thus, the final formula to assess a general compatibility index of two consistent or near consistent vectors *A* and *B* from point to point throughout the both profiles is [1, 2, 5–7]:

$$G(A,B) = \sqrt[1]{2} \sum \left( (ai+bi) \frac{\text{Min}(ai,bi)}{\text{Max}(ai,bi)} \right)^2$$

$$\text{With}: \quad \Sigma a\_i = \Sigma b\_i = \ 1$$

*G AB* ( ) , = general compatibility index of DM1 with respect of DM2.

This can be shown graphically in **Figure 4** [1, 2].

Measuring in Weighted Environments: Moving from Metric to Order Topology (Knowing When Close Really Means Close) http://dx.doi.org/10.5772/63670 255

**Figure 4.** Representation of the cosine projection changing point to point in terms of the profiles.

The *G* function is a transformation function that takes positive real numbers from the range [0, 1], coming from normalized vectors *A* and *B* and returns a positive real number on the same range:

$$a\_i, b\_i X \begin{bmatrix} 0, \ 1 \end{bmatrix} \blacktriangleright G(A, B) \blacktriangleright R^\* X \begin{bmatrix} 0, \ 1 \end{bmatrix}$$

This transformation has two particularly good properties [2]:


It is also possible to define a threshold for compatibility index at 0.901,2. Thus, when two vectors have an index of compatibility equal or greater than 90%, they should be considered compat‐ ible vectors.

Since, incompatibility = 1 – compatibility, then the threshold for tolerable incompatibility is 10%. (This threshold is equivalent with the consistency index in AHP that tolerates a maximum of 10% of inconsistency in the pair comparison matrix.)

Next, a simple application example for 2D vectors *A* and *B* is presented.

The first case is for equal vectors *A* = *B*.

*A* = {0.5; 0.5}; *B* = {0.5; 0.5}, then:

**B.** Two normalized vectors are not closer (not compatible) when the angle (*α*) formed by both vectors on the hyperplane is near 90° or cos *α* is near to 0. From a geometric point of view, they will be represented by perpendicular or nearly perpendicular vectors. In this

**Figure 3** [1, 2] shows the geometric interpretation of vector compatibility. Therefore, there is an operative way to measure compatibility in terms of vector projection. This interpretation of dot product will be very useful for the purposes of finding a compatibility measurement of

Since the space is weighted (we are working in weighted environment), it is also necessary to

weight coordinate by coordinate (coordinate "*i*" may have a different projection and weight than coordinate "*i* + 1"). Thus, the final formula to assess a general compatibility index of two consistent or near consistent vectors *A* and *B* from point to point throughout the both profiles

(,) ½ ( ) Min( , )

With : 1 *i i* S =S = *a b*

*G AB* ( ) , = general compatibility index of DM1 with respect of DM2.

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

) and to take into account the changes of the angle and the

Max( , )

*ai*

*bi*

*ai bi ai bi*

case, they will be defined as noncompatible vectors.

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

**Figure 3.** Geometric interpretation of vector compatibility in terms of its projection.

*G A*

This can be shown graphically in **Figure 4** [1, 2].

weight each projection (each cos *α<sup>i</sup>*

is [1, 2, 5–7]:

two priority vectors in the domain of order topology.

$$G(A,B) \;=\; \forall \left( (0.5 + 0.5)(1/1) + (0.5 + 0.5)(1/1) \right) \;=\; 1.$$

Being 100% compatible or 1–1 = 0% incompatible, this result is expected because *A* and *B* are the same vectors.

The second case is for two very different vectors.

*A*= {0.10; 0.90}; *B* = {0.90; 0.10}, then:

{ } { } { } { } ( ) Min 1; 1 Min 0.1; <sup>1</sup> *a b* == = = = 0.9 0.1; Max 1; 1 *a b* Max 0.1; 0.9 0.9, and Min / Max 0.111.

The same for coordinate 2:

(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*).
