3 Analytical Hierarchy Process (AHP)
21
as well as the consistency index, which will be in Eigenvalue form (denoted as λ).
The mathematical methodology used here is based on the Eigenvalue problem. The
Eigenvector of alternative weights as well as the consistency index is then utilized
to rank the alternatives, thus clarifying the optimal alternative.
When faced with a choice between x number of choices, one can apply the AHP
method to select the optimal choice. To do this, the decision maker must first design
the hierarchy model. This requires an assessment of the problem at hand by categorizing the alternatives together, the criteria and finally, setting a goal. They are
then put into a hierarchy network similar to that shown in Fig. 3.1. Secondly, the
decision-maker will use pairwise comparisons to compare each choice with the others
using the ratio scale of relative importance illustrated in Table 3.1. These comparison
values will be used to construct the pairwise comparison matrix A.
A =
a i j
This matrix will have the following characteristics:
• Square matrix with dimensions of x ×x, where x represents the number of choices
(criteria)
• Leading diagonal input elements will all hold a value of 1.
• Positive matrix; meaning all elements will be greater than 0, non-negative.
a ij > 0 fori, j = 1, . . . , n
• Reciprocal matrix; meaning adjacent inputs will be the reciprocal values of each
other.
a i j =
1
a ji
for i, j = 1, . . . , n
Consistency in judgments is a mandatory concept in AHP that will be thoroughly
discussed in this chapter. To generalize, it legitimizes the applicability of the AHP
method depending on how consistent the decision-maker is when applying his/her
personally motivated preference in judgements. If the following holds true, then the
decision maker is said to be consistent:
a ik = a
∗
ij a kj for i, j, k = 1, . . . , n
In AHP, inconsistency is expected and accounted for. This is because the numerical
values are derived from the decision maker’s preference or individual opinions. In
real life, these values can be inconsistent and therefore these inconsistencies must
be accounted for. To summarize, below is a step-by-step simplified overview of
21
as well as the consistency index, which will be in Eigenvalue form (denoted as λ).
The mathematical methodology used here is based on the Eigenvalue problem. The
Eigenvector of alternative weights as well as the consistency index is then utilized
to rank the alternatives, thus clarifying the optimal alternative.
When faced with a choice between x number of choices, one can apply the AHP
method to select the optimal choice. To do this, the decision maker must first design
the hierarchy model. This requires an assessment of the problem at hand by categorizing the alternatives together, the criteria and finally, setting a goal. They are
then put into a hierarchy network similar to that shown in Fig. 3.1. Secondly, the
decision-maker will use pairwise comparisons to compare each choice with the others
using the ratio scale of relative importance illustrated in Table 3.1. These comparison
values will be used to construct the pairwise comparison matrix A.
A =
a i j
This matrix will have the following characteristics:
• Square matrix with dimensions of x ×x, where x represents the number of choices
(criteria)
• Leading diagonal input elements will all hold a value of 1.
• Positive matrix; meaning all elements will be greater than 0, non-negative.
a ij > 0 fori, j = 1, . . . , n
• Reciprocal matrix; meaning adjacent inputs will be the reciprocal values of each
other.
a i j =
1
a ji
for i, j = 1, . . . , n
Consistency in judgments is a mandatory concept in AHP that will be thoroughly
discussed in this chapter. To generalize, it legitimizes the applicability of the AHP
method depending on how consistent the decision-maker is when applying his/her
personally motivated preference in judgements. If the following holds true, then the
decision maker is said to be consistent:
a ik = a
∗
ij a kj for i, j, k = 1, . . . , n
In AHP, inconsistency is expected and accounted for. This is because the numerical
values are derived from the decision maker’s preference or individual opinions. In
real life, these values can be inconsistent and therefore these inconsistencies must
be accounted for. To summarize, below is a step-by-step simplified overview of