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importance of criteria. If a criterion is less important than other, then the inverse
preference is rated in the ratio scale 1/1, 1/2, 1/3, …, 1/9.
The reason for Satty's scale is based on psychological theories [20] and
experiments that points to the use of nine unit scales as a reasonable set that allows
humans to perform discrimination between preferences for two items. Each value of
the scale can be given a different interpretation allowing a numerical, verbal or
graphical interpretation of the values [20]. An example of pairwise priorities
assignment for a problem with three alternatives is depicted in Table 3. An example
of pairwise comparison to express the relative importance (weights) between two
criteria is depicted in Table 4.
Table 3. Example of AHP pairwise classification of alternatives for a criterion
Criterion
Alternative 1
Alternative
2
Alternative
3
Alternative 1
1
3
9
Alternative 2
1/3
1
5
Alternative 3
1/9
1/5
1
The comparative matrices are reciprocal, hence they have to satisfy the property:
aji=1/aij.
Table 4. Example of AHP pairwise assignment of weights (importance) between two criteria
Weights
Criterion 1
Criterion 2
Criterion 1
1
5
Criterion 2
1/5
1
The interpretation of the matrix in Table 3 is: Alternative 1 is weakly more
important than Alternative 2 and absolutely more important than Alternative 3;
Alternative 2 is moderately more important than Alternative 3. The interpretation for
Table 4 is: the weight (importance) of Criterion 1 is moderately more important than
Criterion 2. Note that the ratios on both matrices (Tables 3, 4) represent the inverse of
the mentioned interpretations, the diagonal is always 1 to express neutrality between
the same alternative or criterion, and the complete matrices are denoted as reciprocal
matrices [20].
Step 4: Synthesis: In this step we calculate the priority of each alternative solution
and criteria being compared. Several mathematical procedures can be used for
synthesization, such as eigenvalues and eigenvectors [20]. However, here, as
mentioned before, we use an averaging method, which provides similar results and is
much simpler to apply [24]. The synthesis is determined in two phases:
1. The process for synthesis follows a bottom-up approach. Hence, starting from
the bottom of the hierarchical tree and using the reciprocal matrices as shown
in Table 3 they are normalized per column and then we calculate the average
per line to obtain a priority vector for each matrix corresponding to each
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