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Prioritization of biofuels production pathways under uncertainties 343
Definition 6 The possibility of an interval being greater than another
+ +
and y¼[y ,y ], the possi-
For two arbitrary interval numbers x ¼ x , x½
bilities that x y and that b a are defined in the following two equations,
respectively (Xu and Da, 2003):
+
y x
+ +
y , y Þ ¼ max 1 max ,0
y y + x x
p xy ¼ px , x½ð ½ + +
(12.7)
2.2 Interval AHP
The interval analytic hierarchy process (IAHP) developed by Xu and Da
(2003) has been widely used in various fields for its advantage of address
ambiguity and hesitation existing in human judgments. Ren (2018) summa-
rized the IAHP method developed by Xu and Da (2003) in the following
three steps:
Step 1: Establishing the interval pair-wise comparison matrix.
Assuming that there are n criteria (C 1 ,C 2 , …,C n ) which need the
stakeholders/decision-makers to determine the relative importance
(weights), the stakeholders/decision-makers were asked to use the Saaty’s
nine-scale system (Saaty, 2008) to establish the pair-wise comparison matrix
(see Table 12.1). However, it is different from the traditional AHP method
which relies on employing the numbers(from 1 to 9) and their correspond-
ing reciprocals to establish the pair-wise comparison matrix, the users of the
interval numbers rather than the single numbers which sometime cannot
Table 12.1 Saaty scales for establishing the pair-wise comparison matrix (Saaty, 2008)
Scales Definition Explanation
1 Equal Two elements perform equally
importance
3 Moderate Experience and judgment slightly favor one element
importance over another
5 Essential Experience and judgment strongly favor one
importance element over another
7 Very strong An element is favored very strongly over another; its
importance dominance demonstrated in practice
9 Absolute The evidence favoring one element over another is
importance of the highest possible order of affirmation
2,4,6,8 Intermediate Intermediate value
value
