# The method of analyzing hierarchies T. Saati - Theory...

## Tahiti Saati's hierarchy analysis method

The hierarchy analysis method was proposed in the late 1970s. American mathematician T. Saati [72]. The method consists in decomposing the problem into simpler parts and stepwise setting the priorities of the evaluated components using pairwise (pairwise) comparisons.

At the first stage, the most important elements of the problem are identified.

The second is the best way to test observations, test and evaluate elements.

On the third stage, a method is developed for applying the solution and assessing its quality.

The entire process is subject to verification and re-evaluation, until it is certain that the process has covered all the important characteristics necessary to present and solve the problem.

The process can be performed on a sequence of hierarchies. In this case, the results obtained in one of them are used as input data in the study of the following.

In the simplest hierarchy, called Saati , it defines three levels: the upper level of the goal (or goals), the middle one - the criteria, the lower one - the list of alternatives (Figure 6.1).

There may be several intermediate levels between the goal and the alternatives (Figure 6.2). For example, the level of problems, actors (the level of operating forces, which may be the administrative authorities, residents, etc.). Each of the criteria can be divided into sub-criteria.

A hierarchy is considered complete if every element of a given level functions as a criterion for all elements of the underlying level. A hierarchy can be divided into sub-hierarchies.

Links between levels are often represented as shown in Fig. 6.2.

To implement the method, the law of hierarchical continuity is introduced, according to which it is required that the elements of each level are comparable to the elements of the higher level.

Matrices are built between levels. For the structure shown in Fig. 6.1, the matrices are constructed as follows: one matrix for comparing the relative importance of the criteria with respect to the target and the matrix for assessing the relative importance of alternatives with respect to each of the second-level criteria. The number of matrices between the level of criteria and alternatives is equal to the number of criteria. The total number of matrices is equal to the number of criteria plus one for evaluating criteria relative to the goal.

Paired comparisons used in the Saati method lead to square matrices of the form

(6.2)

This matrix has the property of inverse symmetry, i.e.

where the indices i and j refer to the row and the column, respectively. The inverse numbers are used later in the processing of the matrix.

Fig. 6.1

Fig. 6.2

In matrices, the elements of the lower level (alternatives, variants) are compared in pairs relative to the criteria, and the criteria are compared to the goal.

These estimates can be obtained in various ways. But in the Saati method, a special scale of 1 to 9 is recommended for the evaluation of components, in which the components of equal importance are assigned a unit, with moderate superiority of 3, with significant superiority of 5, significant superiority of 7, and very strong superiority of 9. Values ​​of 2 , 4.6, 8 are used as intermediate between two neighboring components, which received the estimates 1, 3, 5, 7, 9, respectively.

The relative importance of any element that is comparable to itself is one; the diagonal of the matrix consists of ones. When the matrix is ​​filled, the inverse symmetry property is used: symmetric cells are filled with inverse quantities.

After receiving a set of matrices, you can make a decision based on their meaningful analysis, presenting the decision-maker with an assessment of the alternatives according to the criteria considered. However, it is desirable to obtain generalized estimates of alternatives. For this, different averaging methods can be applied. Saati suggests using the mean geometric averaging and normalization of the generalized estimates obtained. An example of such a procedure is given in Table. 6.1.

Since such approximate calculations of the roots are unavoidable with a sufficiently complicated procedure of evaluating the estimates (especially for a large number of criteria), then to check the consistency of the results obtained, it is suggested to multiply the matrix by normalized estimates and obtain a measure of the degree of deviation from the agreed estimates-the consistency indices for each of the matrices and the hierarchy as a whole:

(6.3)

Table 6.1

 Matrix Computing the estimates of the components of an eigenvector by rows Summarizing column elements and rationing Normalizing results to obtain priority vector estimates Amount

The possibility and expediency of such an assessment is based on the fact that in the above procedure, is nothing more than respectively. However, approximate calculations can lead to misalignment of estimates.

It is also important to note that there are no fractional relations in the matrix of judgments, there are only integers or their reciprocals.

Once the consistency indices are obtained, they are compared to allowances (deviation of 10% or less). If the necessary consistency does not work out, the evaluation procedure must be repeated.

Specific examples of calculations can be found in G. Saati and To. Cairns Analytical Planning: Organization Systems [72, p. 33-43].

When applying the method, it should be borne in mind that, as noted by Saati, complex mathematics can not improve what an individual does not want to change [72, p. 37]. If the required consistency is not obtained, you should return to the survey, changing the wording of the questions, and, if necessary, the criteria. Says Saati also the appropriateness of taking Miller's hypothesis into account: to estimate no more than 7 + 2 components at each level.

## thematic pictures

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