8 Fuzzy Logic and Fuzzy Based Multi Criteria Decision Analysis
55
(c) Weighted Average Method: In this procedure, the membership function is
weighted with its maximum membership degree. The following equation
expresses the weighted average method technique, respectively.
x
∗
=
∫ μ ˜
A (x) · xdx
∫ μ ˜
A (x) · dx
(29)
(d) Mean-Max Membership: This method can be determined as the middle of the
maxima. The output x
∗ is expressed by the formula below:
x
∗
=
n
i=1 ¯
x i
n
(30)
8.5 Fuzzy Sets Algebraic Operations
The main fuzzy algebraic operations used can be summarized as follows
(a) Algebraic Product:
˜
A · ˜
B ⇔ μ ˜
A· ˜
B = μ ˜
A · μ ˜
B
(32)
(b) Algebraic Sum:
˜
A + ˜
B ⇔ μ ˜
A+ ˜
B = μ ˜
A + μ ˜
B − μ ˜
A · μ ˜
B
(33)
(c) Bounded-Sum:
˜
A ⊕ ˜
B ⇔ μ ˜
A⊕ ˜
B = 1 ∧ (μ ˜
A + μ ˜
B )
(34)
(d) Bounded-Difference:
˜
A ˜
B ⇔ μ ˜
A ˜
B = 0 ∨ (μ ˜
A − μ ˜
B )
(35)