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342 Biofuels for a More Sustainable Future

if 0 x x + [see the work of Zhang et al. (2005)]. It is apparent that an

interval number can take an arbitrary value between its lower bound and
upper bound.

Definition 1 Distance between two interval numbers (Yue, 2011)
+ +
Š and y¼[y ,y ] are two arbitrary interval numbers, the dis-

If x ¼ x , x½
+ +

Š to y¼[y ,y ] can be determined by
tance from x ¼ x , x½
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 + + 2
(12.1)
j x yj ¼ ð x y Þ + x yð Þ
Definition 2 Number product between a positive real number and an
interval number (Zhang et al., 2005)
The number product of a positive real number λ and an interval number
+
Š is defined as
x ¼ x , x½
+ +
(12.2)
½
λ x ¼ λ x , x½ Š ¼ λx , λx Š
Note that a crisp number λ could also be transformed into an interval
number λ¼λ,λ].
Definition 3 Addition (Bohlender and Kulisch, 2011)
+ +

Š and y¼[y ,y ] are two arbitrary interval numbers, the sum
If x ¼ x , x½
of the two interval numbers can be obtained by
+ + + +
(12.3)
½
½
x + y ¼ x , x½ Š + y , y Š ¼ x + y , x + y Š
Definition 4 Subtraction (Moore, 1966; Dymova et al., 2013)
+ +

Š and y¼[y ,y ] are two arbitrary interval numbers, the sub-
If x ¼ x , x½
traction between two interval numbers can be determined by
+ + + +
(12.4)
x y ¼ x , x½ Š y , y Š ¼ x y ;x y Š
½
½
Definition 5 Multiplication (Zhang et al., 2005)
+
+
If x ¼ x , x Š and y¼[y ,y ] are two arbitrary interval numbers, the inter-

½
val product can be obtained according to the following two cases:
+
(1) when y >0, the interval product can be determined by
+ + +
(12.5)
½
x y ¼ x , x½ Š y , yŠ ¼ x y , x y Š
½
+
(2) when y <0, the interval product can be determined by
+ + +
+ (12.6)
½
x x ¼ x , x½ Š y , y Š ¼ x y , x y Š
½