EL PROBLEMA DE INVESTIGACIÓN
2.2 Finalidad y Objetivos de la Investigación 1 Finalidad
A fuzzy IF-THEN rulebase is the core of the fuzzy inference process for formulating the mapping from given input fuzzy sets to an output fuzzy set. Once the rulebase is established, fuzzy inference process can be carried out as shown in Fig. 3-3. Fuzzy set inputs are directly input into the fuzzy inference system to determine which rules are
CHAPTER 3: FUNDAMENTALS OF FRA AND MODIFIED FUZZY-AHP
relevant to the current situation, and the results from inference of individual rules are then aggregated to the output result from inference with current inputs. The overall process is developed on the basis of the Mamdani method (Lee, 2005; An et al., 2006 & 2007).
Fig. 3-3 Fuzzy inference process
Supposing there aremrules in the rulebase, and the ith rule is defined as:
1 1
: IF Ri X is Ai and and Xjis Aij and and Xnis Ani , THEN is Y Bi. Eq. 3-10
where there are n arguments in the IF part connected with the AND operation, and
1, 2, , n
X X X and Y are linguistic variables in their corresponding universe discourse, and 1i, 2i i
n
A A A and i
B are linguistic terms of linguistic variables 1, 2, , n
X X X and Y. The calculation of the fire strength i of rule Ri with input fuzzy setsX X1, 2, ,Xn using fuzzy intersection operation is given by:
1
2
1 2
1 1 2 2
min max i , max i , , max n i
n
i X x A x X x A x X xn A xn
CHAPTER 3: FUNDAMENTALS OF FRA AND MODIFIED FUZZY-AHP
Where
1 1 , 2 2 , , n
X x X x X xn
are the MFs of fuzzy sets X X1, 2, ,Xn ,
respectively, and
1 1 , 2 2 , , n
A x A x A xn
are the MFs of fuzzy sets of linguistic terms A A1i, 2i, ,Ani in rule Ri. After the fuzzy implication, the truncated MF Bi of
the inferred conclusion fuzzy set of ruleRi can be obtained by:
i i i
B y B y
Eq. 3-12
Where i is the fire strength of rule Ri, and Bi
y is the MF of linguistic termi
B
andyis an input variable in the universe of discourse.
The firing strength is implicated with the value of the conclusion MF and the output is a truncated MF. The truncated MF's corresponding fuzzy sets that represent the implication output fuzzy sets of rules are aggregated into a single fuzzy set. The MF
B y
of output fuzzy set after aggregation using fuzzy union (maximum) operation is denoted by:
1 V i n B y i B y Eq. 3-13whereBiis the MF of conclusion fuzzy set of ruleRiand n is the total number of rules
in the rule base.
As the output from the fuzzy inference system is a fuzzy set, defuzzification needs to be applied to convert the fuzzy result into a matching numerical value. The centre of area method (Lee, 2005; An et al., 2006 & 2007) is employed for defuzzification. Assume the output fuzzy set obtained from the fuzzy inference system is
, B | , B 0,1
CHAPTER 3: FUNDAMENTALS OF FRA AND MODIFIED FUZZY-AHP
B y
of the conclusion fuzzy set can be calculated by:
1 1 m B j j j m B j j y y c y
Eq. 3-14where m is the number of quantisation levels of the conclusion fuzzy set.
Fig. 3-4 demonstrates the fuzzy inference process with three input fuzzy sets: trapezoidal fuzzy set X1, triangular fuzzy set X2 and crisp set X3. These inputs are fired with Rules Ri and Rj, where Rule Rj is fired twice as fuzzy sets X1 and
1
j
A have two intersection points. The fuzzy sets Y Yi , j, and Yk derived from implication are determined by Minimum operation, and then they are aggregated together by Maximum operation. Finally, the output from the defuzzification is calculated by the centre of area method as described above.
CHAPTER 3: FUNDAMENTALS OF FRA AND MODIFIED FUZZY-AHP
Fig. 3-4 Fuzzy inference process with three input fuzzy sets
FRA is capable to deal with qualitative information so that imprecision or approximate information can be taken into consideration in the risk assessment. It also provides a framework that transforms a human knowledge base into a non-linear mapping. Because the risk contributions of components and subsystems to a system are different, the weight factor (WF) of each component or subsystem within the system should be considered in order to calculate their relative contributions to the RL of the system. Thus, an improved fuzzy analytical hierarchy process technique has been developed and incorporated into the proposed risk model in this study to use its advantages in determining the relative importance of components and subsystems. Therefore, the risk assessment can be progressed from component level to the subsystem level and finally to the system level. Such a technique will be introduced in the following section.
CHAPTER 3: FUNDAMENTALS OF FRA AND MODIFIED FUZZY-AHP