The degree of uncertainty is usually represented by a crisp numerical value on a scale from 0 to 1, where a certainty factor of 1 indicates that the assessment of a particular fact is ve
Trang 1These modifiers change the shape of a fuzzy set using mathematical operations
on each point of the set In the above table, the variable y represents each member-ship value in the fuzzy set, and A represents the entire fuzzy set (i.e the term very A applies the very modifier to the entire set where the modifier description y ∗∗2 squares
each membership value) When a modifier is used in descriptive expressions, it can
be used in upper or lower case (i.e NOT or not)
c) Uncertainty
Uncertainty occurs when one is not absolutely sure about an element of informa-tion The degree of uncertainty is usually represented by a crisp numerical value on
a scale from 0 to 1, where a certainty factor of 1 indicates that the assessment of
a particular fact is very certain that the fact is true, and a certainty factor of 0 indi-cates that the assessment is very uncertain that the fact is true A fact is composed of
two parts: the statement of the fact in non-fuzzy reasoning, and its certainty factor Only facts have associated certainty factors In general, a factual statement takes the following form:
(fact) {CF certainty factor}
The CF acts as the delimiter between the fact and the numerical certainty factor, and the brackets { } indicate an optional part of the statement For example, (pressure high) {CF 0.8} is a fact that indicates a particular system attribute of pressure will be
high with a certainty of 0.8 However, if the certainty factor is omitted, as in a non-fuzzy fact, (pressure high), then the assumption is that the pressure will be high with
a certainty of 1 (or 100%) The term high in itself is fuzzy and relates to a fuzzy set The fuzzy term high also has a certainty qualification through its certainty factor.
Thus, uncertainty and fuzziness can occur simultaneously
d) Fuzzy Inference
Expression of fuzzy knowledge is primarily through the use of fuzzy rules However,
there is no unique type of fuzzy knowledge, nor is there only one kind of fuzzy rule
It is pointed out that the interpretation of a fuzzy rule dictates the way the fuzzy rule should be combined in the framework of fuzzy sets and possibility theory (Dubois
et al 1994)
The various kinds of fuzzy rules that can be considered (certainty rules, gradual
rules, possibility rules, etc.) have different fuzzy inference behaviours, and
corre-spond to various applications Rule evaluation depends on a number of different factors, such as whether or not fuzzy variables are found in the antecedent or conse-quent part of a rule, whether a rule contains multiple antecedents or conseconse-quents, or whether a fuzzy fact being asserted has the same fuzzy variable as an already exist-ing fuzzy fact (global contribution) The representation of fuzzy knowledge through fuzzy inference needs to be briefly investigated for inclusion in engineering design analysis
Trang 2e) Simple Fuzzy Rules
Algorithms for evaluating certainty factors (CF) and simple fuzzy rules are first considered, such as the simple rule of form:
if A then C CFr
f
c where
A is the antecedent of the rule
A is the matching fact in the fact database
C is the consequent of the rule
C is the actual consequent calculated
CFr is the certainty factor of the rule
CFf is the certainty factor of the fact
CFcis the certainty factor of the conclusion
Three types of simple rules are defined:
CRISP_;
FUZZY_CRISP; and
FUZZY_FUZZY
If the antecedent of the rule does not contain a fuzzy object, then the type of rule is CRISP_ regardless of whether or not a consequent contains a fuzzy fact
If only the antecedent contains a fuzzy fact, then the type of rule is FUZZY_CRISP
If both antecedent and consequent contain fuzzy facts, then the type of rule is FUZZY_FUZZY
CRISP_ simple rule If the type of rule is CRISP_, then A must be equal to A in order for this rule to validate (or fire in computer algorithms) This is a non-fuzzy rule (actually, A would be a pattern, and A would match the pattern specification but, for simplicity, patterns are not dealt with here) In this case, the conclusion C
is equal to C, and
FUZZY_CRISP simple rule If the type of rule is FUZZY_CRISP, then A must be
a fuzzy fact with the same fuzzy variable as specified in A for a match In addition, values of the fuzzy variables A and A , as represented by the fuzzy sets Fα and F
α,
do not have to be equal
For a FUZZY_CRISP rule, the conclusion C is equal to C, and
S is a measure of similarity between the fuzzy sets Fα (determined by the fuzzy
pattern A) and F
α (of the matching fact A ) The measure of similarity S is based upon the measure of possibility P and the measure of necessity N It is calculated
Trang 3according to the following formula
S = PFα|F
α
if N
Fα|F
α
> 0.5
S=N
Fα|F
α
+ 0.5∗ PFα|F
α
Otherwise where∀ u ∈ U:
P
Fα|F
α
[min is the minimum and max is the maximum, so that max (min(a,b)) would
represent the maximum of all the minimums between pairs a and b] (Cayrol et al.
1982), and
N (Fα|Fα ) = 1 − PF
α|Fα
(3.97)
F
α is the complement of Fαdescribed by the membership function
∀(u ∈ U)μFα (u) = 1 −μFα(u) (3.98) Therefore, if the similarity between the fuzzy sets associated with the fuzzy
pat-tern (A) and the matching fact (A ) is high, the certainty factor of the conclusion is
very close to CFr∗ CFf , since S will be close to 1 If the fuzzy sets are identical, then S will be 1 and the certainty factor of the conclusion will equal CFr∗ CFf If the match is poor, then this is reflected in a lower certainty factor for the conclusion Note also that if the fuzzy sets do not overlap, then the similarity measure would be zero and the certainty factor of the conclusion would be zero as well In this case, the conclusion would not be asserted and the match considered to have failed, with the outcome that the rule is not to be considered (Orchard 1998)
FUZZY_FUZZY simple rule If the type of rule is FUZZY_FUZZY, and the
fuzzy fact and antecedent fuzzy pattern match in the same manner as discussed for a FUZZY_CRISP rule, then it can be shown that the antecedent and consequent
of such a rule are connected by the fuzzy relation (Zadeh 1973):
where:
Fα = fuzzy set denoting the value of the fuzzy antecedent pattern
Fc = fuzzy set denoting the value of the fuzzy consequent
The membership function of the relation R is calculated according to the following
formula
∀(uv) ∈ U ×V
Trang 4The calculation of the conclusion is based upon the compositional rule of infer-ence, which can be described as follows (Zadeh 1975):
F
c= F
F
c is a fuzzy set denoting the value of the fuzzy object of the consequent The
membership function of F
cis calculated as follows (Chiueh 1992):
μF
c(v) = max
u ∈U
minμFα (u) ,μR(u,v)
which may be simplified to
μF
where:
z= maxmin
μFα (u) , μFα(u)
The certainty factor of the conclusion is calculated according to the formula
f) Complex Fuzzy Rules
Complex fuzzy rules—multiple consequents and multiple antecedents—include multiple patterns that are treated as multiple rules with a single assertion in the consequent
Multiple consequents The consequent part of a fuzzy rule may contain only
mul-tiple patterns, specifically (C1, C2, ,Cn), which are treated as multiple rules with
a single consequent Thus, the following rule,
if Antecedents then C1and C2and and C n
is equivalent to the following rules:
if Antecedents then C1
if Antecedents then C2
if Antecedents then C n
Trang 5Multiple Antecedents
From the above, it is clear that only the problem of multiple patterns in the an-tecedent with a single assertion in the consequent needs to be considered If the consequent assertion is not a fuzzy fact, then no special treatment is needed, since the conclusion will be the crisp (non-fuzzy) fact However, if the consequent as-sertion is a fuzzy fact, the fuzzy value is calculated using the following algorithm (Whalen et al 1983)
If the logical term, and, is used:
if A1and A2then C CFr
A
A
c
A
1and A
2are facts (crisp or fuzzy), which match the antecedents A1and A2 respec-tively
In this case, the fuzzy set describing the value of the fuzzy assertion in the con-clusion is calculated according to the formula
F
c = F c1 ∩ F
where∩ denotes the intersection of two fuzzy sets in which a membership function
of a fuzzy set C, which is the intersection of fuzzy sets A and B, is defined by the
following formula
μC(x) = min(μA(x) ,μB(x)) , for x ∈ U (3.105) and:
F
c1is the result of fuzzy inference for the fact A
1and the simple rule:
if A1then C
F
c2is the result of fuzzy inference for the fact A
2and the simple rule:
if A2then C
g) Global Contribution
In non-fuzzy knowledge, a fact is asserted with specific values If the fact already exists, then the approach would be as if the fact was not asserted (unless fact dupli-cation is allowed) In such a crisp system, there is no need to reassess the facts in the system—once they exist, they exist (unless certainty factors are being used, when the certainty factors are modified to account for the new evidence) In a fuzzy sys-tem, however, refinement of a fuzzy fact may be possible Thus, in the case where
Trang 6a fuzzy fact is asserted, this fact is treated as contributing evidence towards the con-clusion about the fuzzy variable (it contributes globally) If information about the fuzzy variable has already been asserted, then this new evidence (or information) about the fuzzy variable is combined with the existing information in the fuzzy fact Thus, the concept of restrictions on fact duplication for fuzzy facts does not apply as
it does for non-fuzzy facts There are many readily identifiable methods of combin-ing evidence In this case, the new value of the fuzzy fact is calculated accordcombin-ingly
where:
Fg is the new value of the fuzzy fact
Ff is the existing value of the fuzzy fact
F
c is the value of the fuzzy fact to be asserted
where∪ denotes the union of two fuzzy sets in which a membership function of
a fuzzy set C, which is the union of fuzzy sets A and B, is defined by the following
formula
The uncertainties are also aggregated to form an overall uncertainty Basically,
two uncertainties are combined, using the following formula
where:
CFg is the combined uncertainty
CFf is the uncertainty of the existing fact
CFc is the uncertainty of the asserted fact
3.3.2.5 Fuzzy Logic and Fuzzy Reasoning
The use of fuzzy logic and fuzzy reasoning methods are becoming more and more popular in intelligent information systems (Ryan et al 1994; Yen et al 1995), in knowledge formation processes within knowledge-based systems (Walden et al 1995), in hyper-knowledge support systems (Carlsson et al 1995a,b,c), and in active decision support systems (Brännback et al 1997)
a) Linguistic Variables
As indicated in Sect 3.3.2.4, the use of fuzzy sets provides a basis for the manipula-tion of vague and imprecise concepts Fuzzy sets were introduced by Zadeh (1975)
as a means of representing and manipulating imprecise data and, in particular, fuzzy
Trang 7sets can be used to represent linguistic variables A linguistic variable can be
re-garded either as a variable of which the value is a fuzzy number or as a variable
of which the values are defined in linguistic terms, such as failure modes, failure effects, failure consequences and failure causes in FMEA and FMECA.
A linguistic variable is characterised by a quintuple
where:
x is the name of the linguistic variable;
T (x) is the term set of x, i.e the set of names of linguistic values
of x with each value being a fuzzy number defined on U ;
G is a syntactic rule for generating the names of values of x;
M is a semantic rule for associating with each value its meaning
Consider the example If pressure in a process design is interpreted as a linguistic
variable, then its term set T (pressure) could be: T = {very low, low, moderate,
high, very high, more or less high, slightly high, } where each of the terms in
T (pressure) is characterised by the fuzzy set in a universe of discourse U = [0,300]
with a unit of measure that the variable pressure might have.
We might interpret:
low as ‘a pressure below about 50 psi’
moderate as ‘a pressure close to 120 psi’
high as ‘a pressure close to 190 psi’
very high as ‘a pressure above about 260 psi’
These terms can be characterised as fuzzy sets of which the membership functions
are:
1− (p − 50)/70 if 50≤ p ≤ 120
moderate(p) =
1− |p − 120|/140 if 50 ≤ p ≤ 190
high(p) =
1− |p − 190|/140 if 120 ≤ p ≤ 260
very high(p) =
1− (260 − p)/140 if 190 ≤ p ≤ 260
The term set T (pressure) given by the above linguistic variables, T (pressure) =
{low (p), moderate (p), high (p), very high (p)}, and the related fuzzy sets can be represented by the mapping illustrated in Fig 3.29.
Trang 8very high high
moderate low
Fig 3.29 Values of linguistic variable pressure
A mapping can be formulated as:
T : [0,1] × [0,1] → [0,1]
which is a triangular norm (t-norm for short) if it is symmetric, associative and
non-decreasing in each argument, and T (a,1) = a, for all a ∈ [0,1].
The mapping formulated by
S : [0,1] × [0,1] → [0,1]
is a triangular co-norm (t-conorm, for short) if it is symmetric, associative and
non-decreasing in each argument, and S (a,0) = a, for all a ∈ [0,1].
b) Translation Rules
Zadeh introduced a number of translation rules that allow for the representation of common linguistic statements in terms of propositions (or premises) These
transla-tion rules are expressed as (Zadeh 1979):
Main premise
Helping premise
Conclusion
x is A
x is B
x is A ∩ B
x is an element of set A
x is an element of set B
x is an element of intersection A and B
Some of the translation rules include:
Entailment rule:
x is A
A ⊂ B
x is B
pressure is very low very low⊂ low
pressure is low
Conjunction rule:
x is A
x is B
x is A ∩ B
pressure is not very high pressure is not very low
pressure is not very high and not very low
Trang 9Disjunction rule:
x is A
or x is B
x is A ∪ B
pressure is not very high
or pressure is not very low
pressure is not very high or not very low
Projection rule: (x, y) have relation R
x is∏X (R)
(x, y) have relation R
y is∏Y (R)
where:∏X is a possibility measure defined on a finite propositional language
and R is a particular rule-base (defined later).
Negation rule: not (x is A)
x is ¬A
not (x is high)
x is not high
c) Fuzzy Logic
Prior to reviewing fuzzy logic, some consideration must first be given to crisp logic, especially on the concept of implication, in order to understand the comparable con-cept in fuzzy logic Rules are a form of propositions A proposition is an ordinary
statement involving terms that have been defined, e.g ‘the failure rate is low’ Con-sequently, the following rule can be stated: ‘IF the failure rate is low, THEN the equipment’s reliability can be assumed to be high’
In traditional propositional logic, a proposition must be meaningful to call it
‘true’ or ‘false’, whether or not we know which of these terms properly applies
Logical reasoning is the process of combining given propositions into other
propo-sitions, and repeating this step over and over again Propositions can be com-bined in many ways, all of which are derived from several fundamental operations (Bezdek 1993):
• conjunction denoted p∧q where we assert the simultaneous truth of two separate propositions p and q;
• disjunction denoted p ∨ q where we assert the truth of either or both of two
sep-arate propositions; and
• implication denoted p → q, which takes the form of an IF–THEN rule The IF part of an implication is called the antecedent, and the THEN part is called the consequent.
• negation denoted by (∼p) where a new proposition can be obtained from a given
one by the clause ‘it is false that ’
• equivalence denoted by p ↔ q, which means that p and q are both true or false.
In traditional propositional logic, unrelated propositions are combined into an
impli-cation, and no cause or effect relation is assumed to exist This results in
fundamen-tal problems when traditional propositional logic is applied to engineering design analysis, such as in a diagnostic FMECA, where cause and effect are definite (i.e causes and effects do occur)
Trang 10In traditional propositional logic, an implication is said to be true if one of the
following holds:
1) (antecedent is true, consequent is true),
2) (antecedent is false, consequent is false),
3) (antecedent is false, consequent is true)
The implication is said to be false when:
4) (antecedent is true, consequent is false)
Situation 1 is familiar from common experience Situation 2 is also reasonable be-cause, if we start from a false assumption, then we expect to reach a false conclusion However, intuition is not always reliable We may reason correctly from a false an-tecedent to a true consequent Hence, a false anan-tecedent can lead to a consequent that is either true or false, and thus both situations 2 and 3 are acceptable in tradi-tional propositradi-tional logic Finally, situation 4 is in accordance with intuition, for an implication is clearly false if a true antecedent leads to a false consequent
A logical structure is constructed by applying the above four operations to
propo-sitions The objective of a logical structure is to determine the truth or falsehood
of all propositions that can be stated in the terminology of this structure A truth table is very convenient for showing relationships between several propositions The fundamental truth tables for conjunction, disjunction, implication, equivalence and negation are collected together in Table 3.14, in which symbol T means that the
corresponding proposition is true, and symbol F means it is false The fundamental axioms of traditional propositional logic are:
1) Every proposition is either true or false, but not both true and false
2) The expressions given by defined terms are propositions
3) Conjunction, disjunction, implication, equivalence and negation
Using truth tables, many interpretations of the preceding translation rules can be derived
A tautology is a proposition formed by combining other propositions, which is
true regardless of the truth or falsehood of the forming propositions The most im-portant tautologies are
These tautologies can be verified by substituting all the possible combinations
for p and q and verifying how the equivalence always holds true The importance of these tautologies is that they express the membership function for p → q in terms of membership functions of either propositions p and ∼q or ∼p and q, thus giving the
following