3.3 Analytic Development of Reliability and Performance in Engineering Design 153 These 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 appliesthe very modifierto theentire set wherethe modifierdescriptiony ∗∗ 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 rep resented 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 certain ty factor of 0 indi- cates that the assessment is very uncertain that the fact is true. A fact is composed of two p arts: 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 p ressure 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 Expressionof 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 consequents, o r 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. 154 3 Reliability and Performance in Engineering Design e) 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 CF r A  CF f C  CF 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 CF r is the certain ty factor of the rule CF f is the certainty factor of the fact CF c is the certainty factor of the con clusion 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-fu zzy 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 CF c = CF r ∗CF f . (3.94) FUZZY_CRISP simple rule If the type of rule is FUZZY_CRISP, then A  must be a f uzzy 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 F α and F  α , do not have to be equal. For a FUZZY_CRISP rule, the conclusion C  is equal to C,and CF c = CF r ∗CF f ∗S . (3.95) S is a measure of similarity between the fuzzy sets F α (determined by the fuzzy pattern A)andF  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 3.3 Analytic Development of Reliability and Performance in Engineering Design 155 according to the following formula S = P  F α |F  α  if N  F α |F  α  > 0.5 S =  N  F α |F  α  + 0.5  ∗P  F α |F  α  Otherwise where ∀ u ∈ U: P  F α |F  α  =max  min  μ F α (u) , μ F  α (u)  (3.96) [min is the minim um and max is th e maximum, so that max (min(a,b)) would represent the maximum of all the m inimums between pairs a and b ] (Cayrol et al. 1982), and N (F α |F α  )=1−P  F  α |F α   (3.97) F  α is the complement of F α described by the membership function ∀(u ∈U) μ F  α (u)=1− μ F α (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 CF r ∗CF f ,sinceS will be close to 1. If the fuzzy sets are identical, then S will be 1 and the certainty factor of the c onclusion will equal CF r ∗CF f .If the match is poor, then this is r eflected 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): R = F α ∗F c (3.99) where: F α = fuzzy set denoting the value of the fuzzy antecedent pattern F c = fuzzy set denoting the value of the fuzzy consequent The membership function of the relation R is calculated according to the following formula μ R(u,v)=min( μ F α (u) , μ F c (v)) , (3.100) ∀(uv) ∈U ×V 156 3 Reliability and Performance in Engineering Design The calculation of the conclusion is based upon the compositional rule of infer- ence, which can be described as follows (Zadeh 1975): F  c = F  α ◦ R (3.101) F  c is a fuzzy set denoting the value of the fuzzy object of the consequent. The membership function of F  c is calculated as follows (Chiueh 1992): μ F  c (v)=max u∈U  min μ F α  (u) , μ R (u,v)  which may be simplified to μ F  c (v)=min(z, μ F c (v)) (3.102) where: z = max  min  μ F α  (u) , μ F α (u)  The certainty factor of the conclusion is calculated according to the formula CF c = CF r ∗CF f (3.103) 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 (C 1 , C 2 , ,C n ), which are treated as multiple rules with a single consequent. Thus, the following rule, if Antecedents then C 1 and C 2 and . and C n is equivalent to the following rules: if Antecedents then C 1 if Antecedents then C 2 if Antecedents then C n 3.3 Analytic Development of Reliability and Performance in Engineering Design 157 Multiple Anteceden ts From the above, it is clear that only the problem of multiple pattern s 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, th e fuzzy value is calculated using the following algorithm (Whalen et al. 1983). If the logical term, and,isused: if A 1 and A 2 then C CF r A  1 CF f1 A  2 CF f2 C  CF c A  1 and A  2 are facts (crisp o r fuzzy), which match the antecedents A 1 and A 2 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  c2 (3.104) where ∩ denotes the intersection of two fu zzy sets in which a m embership function of a fuzzy set C, which is the intersection of fuzzy sets A and B, is defined by the following formula μ C (x)=min( μ A (x) , μ B (x)) , for x ∈U (3.105) and: F  c1 is the resu lt of fuzzy infer ence for the fact A  1 and the simple rule: if A 1 then C F  c2 is the resu lt of fuzzy infer ence for the fact A  2 and the simple rule: if A 2 then C g) Global Contribution In non-fuzzy knowledge, a fact is asserted with specific values. If the fact already exists, then the approach would b e 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 158 3 Reliability and Performance in Engineering Design a 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 n ew 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 o f the fuzzy fact is calculated accordingly F g = F f ∪F  c (3.106) where: F g is the new value of the fuzzy fact F f is th e 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 μ C (x)=max( μ A (x) , μ B (x)) for x ∈U (3.107) The uncertainties are also aggregated to form an overall uncertainty. Basically, two uncertainties are combined, using the following formula CF g = maximum(CF f ,CF c ) (3.108) where: CF g is the combined uncertainty CF f is the uncertainty of the existing fact CF c 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-knowledgesupportsystems (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 3.3 Analytic Development of Reliability and Performance in Engineering Design 159 sets can be used to represent linguistic variables. A linguistic variable can be re- garded either as a variable of which th e 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 ca uses in FMEA and FMECA. A linguistic variable is characterised by a quintuple (x,T(x),U,G,M) (3.109) 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 ru le for generating the name s o f 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: low (p)=       1ifp ≤ 50 1−(p −50)/70 if 50 ≤ p ≤ 120 0otherwise moderate (p)=     1−|p −120|/140 if 50 ≤ p ≤ 190 0otherwise high (p)=     1−|p −190|/140 if 120 ≤ p ≤ 260 0otherwise very high (p)=       1ifp ≤ 260 1−(260− p)/140 if 190 ≤ p ≤ 260 0otherwise 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. 160 3 Reliability and Performance in Engineering Design 1 0 50 120 190 260 pressure very highhighmoderatelow 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,foralla ∈[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,foralla ∈ [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 3.3 Analytic Development of Reliability and Performance in Engineering Design 161 Disjunction 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 p articular 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 mu st 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 b e assumed to be high’. In traditional propositional logic, a proposition m ust 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 separa te 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 o f an im plication is called the antecedent, and the THEN p ar t is called the consequent. • negation denoted by (∼p) where a new proposition can be obtained from a given onebytheclause‘itisfalsethat ’. • 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 d o o ccur). 162 3 Reliability and Performance in Engineering Design In 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 antecedent can lead to a consequent that is either true or false, and thus both situations 2 and 3 are acceptable in tradi- tional propositional 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 applyingthe abovefour operations to propo- sitions. The objective o f 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 p ropositions. 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 (p →q) ↔∼[p∧(∼q)] ↔ (∼p) ∨q (3.110) 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 μ p→q (x,y)=1− μ p∩q (x,y)=1−min  μ p (x),1− μ q (y)  (3.111) μ p→q (x,y)= μ p∪q (x,y)=1−max  1− μ p (x), μ q (y)  . (3.112) . investigated for inclusion in engineering design analysis. 154 3 Reliability and Performance in Engineering Design e) Simple Fuzzy Rules Algorithms for evaluating certainty factors ( CF) and simple. representing and manipulating imprecise data and, in particular, fuzzy 3.3 Analytic Development of Reliability and Performance in Engineering Design 159 sets can be used to represent linguistic. Analytic Development of Reliability and Performance in Engineering Design 153 These modifiers change the shape of a fuzzy set using mathematical operations on each point of the set. In the above table,