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The mathematical technique at the core of analytical decision modelling is the Eigen decomposition theorem (Saaty 1990; Vargas 1982), illustrated in Figure 4 with the relationship between a decision matrix of size n and its priority vector (w1...wn).
C = decision factors, wi/w j= relative weights of decision factors Figure 4: Matrix of paired comparisons used in decision making
The fundamental scale of absolute numbers (Saaty 1990) in Table 1 is used to construct pairwise comparison matrices for all level 2 decision factors in Figure 3. Eigen decomposition theorem is then applied to derive the priority vector (v), and the principal eigenvalue (λmax) for each decision matrix.
table 1: fundamental scale of absolute numbers
Intensity of importance
definition
explanation
1
Equal importance
Two activities contribute equally to the objective
2
Weak or slight
3
Moderate importance
Experience and judgement slightly favour one activity over another
4
Moderate plus
5
Strong importance
Experience and judgement strongly favour one activity over another
6
Strong plus
7
Very strong or demonstrated importance
An activity is favoured very strongly over another; its dominance is demonstrated in practice
8
Very, very strong
9
Extreme importance
The evidence favouring one activity over another is of the highest possible order of affirmation.
A reasonable assumption
reciprocals of the above
If activity i has one of the above non-zero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i.
Source: Saaty (2016)
For a perfectly square pairwise comparison matrix, λmax=n........................ (1), where n=size of the matrix (Vargas 2008). The Consistency Index (CI) is determined by the relationship, CI = (λmax-n)/(n-1).......(2) (Saaty 2008a; Franek and Kresta 2014). Consistency measurements based on empirical data shown in Table 2 are then used to determine consistency in decision-making.
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