Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 17 ppt

3.3 Analytic Development of Reliability and Performance in Engineering Design 143 presence of non-linear properties (for example, in the modelling of performance characteristics of relief valves, non-return valves, end stops, etc.). Secondly, the solutions may be very specific. They are typically produced for a system at a certain pressure, flow, load condition, etc. In engineering design, and in particular in the FMEA, it is common not to know the precise values of quantities, especially in the early design stag es. It would thus be more intuitive to be able to relate design criteria in terms of ranges of values, as considered in the labelled interval calculus method for system performance measures. b) Order of Magnitude The problem of how to address complicated failure modes can be approached through order of magnitude reasoning, developed by Raiman (1986) and extended by Mavrovouniotis and Stephanopoulis (Mavrovouniotis et al. 1988). Order of magnitude is primarily concerned with considering the relative sizes of quantities. A variable in this formalismrefers to a specific physicalquantity with known dimen- sions but unknown numerical values. The fundamental concept is that of a link—the ratio of two quantities, only one of which can be a landmark. Such a landmark is a variable with known (and constant) sign and value. There are seven possible prim- itive relations between these two quantities: A << BAis much smaller than B A − < BAis moderately smaller than B A ∼< BAis slightly smaller than B A == BAis exactly equal to B A >∼ BAis slightly larger than B A > − BAis mode rately larger than B A >> BAis much larger than B. The formalism itself involves representing these primitives as real intervals centred around unity (which represents exact equality). They allow the data to be repre- sented either in terms of a precise value or in terms of intervals, depending upon the information available and the problem to be solved. Hence, the algorithmic model will encapsulate all the known features of the system b eing simulated. Vagueness is introduced only by lack of knowledge in the initial conditions. A typical analysis will consist of asking questions of the form: • What happens if the p ressure rises significantly higher than the operating pres- sure? • What is the effect of the flow significantly being reduced? c) Qualitative Simulation Qualitative methods have been devised to simulate physical systems whereby quan- tities are represented by their sig n only, and differential equations are rein terpreted 144 3 Reliability and Performance in Engineering Design as logical predicates. The simulation involves finding values that satisfy these con- straints (de Kleer et al. 1984). This work was further developedto represent the quantities by intervals and land- mark values (Kuipers 1986). Collectively, variables and landmarks are described as the quan tities of the system. The latter represent important values of the quantities such as maximum pressure, temperature, flow, etc. The major drawback with these methods is that the vagueness of the input data leads to ambigu ities in the predictions of system behaviour, whereby many new constraints can be chosen that correspond to many physical solutions. In general, it is not possible to deduce which of the myriad of solutions is correct. In terms of FMEA, this would mean there could be a risk of failure effects being generated that are a result of the inadequacy of the algorithm, and not of a particular failure mode. d) Fuzzy Techniques Kuiper’s work was enhanced by Shen and Leitch (Shen et al. 1993) to allow for fuzzy intervals to be used in fuzzy simulation. In qualitative simulation, it is possible to describe quantities (such as pressure) as ‘low’ or ‘high’. However, typical of engineering systems, these fuzzy intervals may be divided by a land mark representing some critical quantity, with consequent uncertainty where the resulting point should lie, as ‘low’ and ‘high’ are not absolute terms. The concept of fuzzification allows the boundary to be blurred, so that for a small range of values, the quantity could be described as both ‘low’ and ‘medium’. The problem with this approach (and with fuzzy simulation algorithms in general) is that it introduces further amb iguity. For example, it has been found that in the dynamic simulation of an actuator, there are 19 possible values for the solution after only three steps (Bull et al. 1995b). This result is even worse than it appears, as the process of fuzzification removes the guarantee of convergingon a physical solution. Furthermore, it has been shown that it is possible to develop fuzzy Euler integration that allows for qualitative states to be predicted at absolute time poin ts. This solves some of the problems but there is still ambiguity in predicted behaviour of the system (Steele et al. 1996, 1997; Coghill et al. 1999a,b). 3.3.2.3 Qualitative Reasoning in Failure Modes and Effects Analysis It would initially appear that qualitative reasoning algorithms are not suitable for FMEA or FMECA, as this formalism of analysis requires unique predictions of system behaviour. Although some vagueness is permissible due to uncertainty,it cannot be ambiguous, and ambiguity is an inherent feature of computational quali- tative reasoning. I n orde r, then, to consider the feasibility of qualitative reasoning in FMEA and FMECA without this resulting in ambiguity, it is essential to investigate further the concept o f uncertainty in engineering design analysis. 3.3 Analytic Development of Reliability and Performance in Engineering Design 145 a) The Concept of Uncertainty in Engineering Design Analysis Introducing the concept of uncertainty in reliability assessment by utilising the tech- niques of FMEA and FMECA requires that some issues and concepts relating to the physical system being designed must first be considered. A typical engineering design can be defined using the concepts introduced by Simon (1981), in terms of its inner and outer environment, whereby an interface between the substance and organisation of the design itself, and the surroundings in which it o perates is defined. Th e design engineer’s task is to establish a complete definition of the design and, in ma ny cases, the manufacturing details (i.e. the inner environment) that can cope with supply and delivery (i.e. the outer environment) in order to satisfy a predetermined set of design criteria. Many of the issues that are often referred to as uncertainty are related to the ability of the design to meet the design criteria, and are due to characteristics associated with both the innerand outer environments (Batill et al. 2000). This is especially the case when several systems are integrated in a complex process with multiple (often co nflicting) char acteristics. Engineering design is associated with decisions based upon information related to this interface, which considers uncertainty in the complex integration of systems in reality, compared to the concept of uncertainty in systems analysis and modelling. From the perspective of the designer, a primary concern is the source of variations in the inner environment, and the need to reduce these variations in system perfor- mance throughdecisions made in the design process. The designer is also concerned with how to reduce the sensitivity of the system’s performance to variations in the outer environment (Simon 1981). Furthermore, from the designer’s perspective, the system being designed exists only as an abstraction, and any information related to the system’s characteristics or behaviour is approximate prior to its physical reali- sation. Dealing with this incomplete description of the system, and the appr oximate nature of the information associated with its characteristics and behaviour are key issues in the design process (Batill et al. 2000). The intention, however, is to focus on the integrity of engineering design using the extensive capabilities now available with modelling and digital computing. With the selection of a basic concept of the system at the beginning of the conceptual phase of the engineering design process, the next step is to identify (though not necessarily quantify) a finite set of design variables that will eventually be used to uniquely specify the design. The identification and quantification o f this set of de- sign variables are central to, and will evolve with the design throughout the design process. It is this quantitative description of the system, b ased upon information developed, using algorithmic models or simulation, that becomes the focus of pre- liminary or schematic design. Though there is great benefit in providing quantitative descriptions as early in the design process as possible, this depends upon the availability of knowledge, and the level of analysis and mode lling techniq ues related to the design . As the level of abstraction of the design changes, and more and more detail is required to define it, the number o f design variables will grow consid erably. Design variables typically are associated with the type of material used and the geometric d escription of the 146 3 Reliability and Performance in Engineering Design system(s) beingdesigned.Eventually,duringthe detail design phase ofthe engineer- ing design process, the designer will be required to specify (i.e. quantify) the design variables representing the sy stem. This specification often takes the form of detailed engineering drawings that include materials information and all the necessary geo- metric information needed for fabrication, including manufacturing tolerances. Decisions associated with quantifying (or selecting) the design variables are usu- ally based upon an assessment of a set of behavioural variables, also referred to as system states. The behavioural variables or system states are used to describe the system’s characteristics. The list of these characteristics also increases in detail as the level of abstraction of the system decreases. The behavioural variables are used to assess the suitability of the design, and are based upon information obtained from several primary sources during the design process: • Archived experience • Engineering analysis (such as FMEA and FMECA) • Modelling and simulation. Interpolating or extrapolating fro m information on similar design concepts can pro- vide the designer with sufficient confidence to make a decision based upon the suc- cess of earlier, similar designs. Often, this type of information is incorporated into heuristics (rules-of-thumb),design handbooksor design guidelines. Engineers com- monly gather experiential information from empirical data or knowledge bases. The use of empirical information requires the designer to make numerous assumptions concerning the suitability of the available information and its applicability to the current situation. There are also many decisions made in the design process that are based upon individual or corporate experience that is not formally archived in a database. This type of information is very valuable in the design of systems that are pertur- bations (evolutionary designs) of existing successful designs, but has severe limita- tions when considering the design of new or revolutionary designs. Though it may be useful information, in a way that will assist in assessing the risk associated with the entire design—which is usually not possible, it tends to compound the problem related to the concept of uncertainty in the engineering design process. The second type of information available to the designer is based upon analy- sis, mathematical modelling and simulation. As engineering systems become more complex, and greater demands are placed upon their performance and cost, this source of information becomes even more important in the design process. How- ever, the information provided by analysis such as FMEA and FMECA carries with it a significant level of uncertainty, and the use of such information introduces an equal level of risk to the decisions made, which will affect the integrity of the de- sign. Quantifying uncertainty, and understanding the significant impact it has in the design process, is an important issue that requires specific consideration, especially with respect to the increasing complexity of engineering designs. A further extension to the reliability assessment technique of FMECA is there- fore considered that includes the appropriate representation of uncertainty and 3.3 Analytic Development of Reliability and Performance in Engineering Design 147 incompleteness of information in available knowledge. The main consideration of such an approach is to provide a qualitative treatment of uncertainty based on pos- sibility theory and fuzzy sets (Zadeh 1965). This allows for the realisation of failure effects and overall consequences (manifestations) that will be more or less certainly present (or absent), and failure effects and consequences that could be more or less possibly present (or absent) when a particular failure mode is identified. This is achieved by means of qualitative uncertainty calculus in causal matrices, based on Zadeh’s possibility measures (Zadeh 1979), and their dual measures of certainty (or necessity). b) Uncertainty and Incompleteness in Available Knowledge Available knowledge in engineering design analysis (specifically in the reliability assessment techniques of FMEA and FMECA) can be considered from the point of view of behavioural knowledge and of functional knowledge. These two aspects are accordingly described: i) In behavioural knowledge: expressing the likelihood of some or other expected consequences as a result of an identified failure mode. Information about likeli- hood is generally qualitative, rather than quantitative. Included is the concept of ‘negative information’, stating that some consequences cannot manifest, or are almost impossible as consequences of a hypothesised failure mode. Moreover, due to incompleteness of the knowledge, distinction is m ade between conse- quences that are more or less sure, and those that are only possible. ii) In functional knowledge: expressing the functional activities or work that sys- tems and equipment are designed to perform. In a similar way as in the b e- havioural knowledge, the propagation of system and equipment functions are also incomplete and uncertain. In order to effectivelycapture uncertainty, a qual- itative approach is more approp riate to the available information than a quanti- tative one. In the following paragraphs, an overview is given of various concepts and theory for qu alitatively modelling uncertainty in engineering design. 3.3.2.4 Overview of F uziness in Engineering Design Analysis In the real world there exists knowledge that is vague, uncertain, ambiguous or probabilistic in nature, termed fuzzy knowledge. Human thinking and reasoning fre- quently involves fuzzy knowledge originating from inexact concepts and similar, rather than identical experiences. In complex systems, it is very difficult to answer questions on system behaviour because they generally do not have exact answers. Qualitative reasoning in engineering design analysis attempts n ot only to give such answers but also to describe their reality level, calculated from the uncertainty and imprecisionof facts that are applicable.The analysisshould also be able to copewith 148 3 Reliability and Performance in Engineering Design unreliable and incomplete information and with different expert opinions. Many commercial expert system tools or shells use d ifferent approaches to handle uncer- tainty in knowledge or data, such as certainty factors (Shortliffe 1976) and Bayesian models (Buchanan et al. 1984), but they cannot cope with fuzzy knowledge, which constitutes a very significant part of the use of natural language in design analysis, particularly in the early pha ses of the engineering design pro cess. Several computer automated systems support some fuzzy reasoning, such as FAULT (Whalen et al. 1982), FLOPS (Buckley et al. 1987), FLISP (Sosnowski 1990) and CLIPS (Orchard 1998), though most of these are developed from high- level languages intended for a specific application. Fuzziness and Probability Probability and fu zziness are related but different concepts. Fuzziness is a typ e of deterministic uncertainty. It describes the eventclass ambiguity. Fuzziness measures the degree to which an event occurs, not whether it does occur. Pr obability arises from the question whether or not an event occurs, and assumes that the event class is crisply defined and that the law of non-contradiction holds. However, it would seem more appropriate to investigate the fuzziness of probability, rather than dismiss probability as a sp ecial case of fu zziness. In essence, whenever th e outcome of an event is difficult to compute, a probabilistic approach may be used to estimate the likelihood of all possible outcomes belonging to an event class. Fuzzy probability extends the traditional notion of probability when there are outcomes that belong to several event classes at the same time but at different degrees. Fuzziness and probability are orthogonal concepts that characterise different aspects o f the same event (Bezdek 1993). a) Fuzzy Set Theory Fuzziness occurs when the boundary of an element of information is not clear-cut. For example, concepts such as high, low, medium or even reliable are fuzzy. As a simple example, there is no single quantitative value that defines the term young. For some people, age 25 is young and, for others, age 35 is young. In fact, the concept young has no precise boundary. Age 1 is definitely young and age 100 is definitely not young; however, age 35 has some possibility of being young and usu- ally depends on the context in which it is being considered. The representation of this kind of inexact information is based on the concept of fuzzy set theory (Zadeh 1965). Fuzzy sets are a generalisationof conventional set theory that was introduced as a mathematical way to r epresent vagueness in everyday life. Unlike classical set theory, where one deals with objects of which the membership to a set can be clearly described, in fuzzy set theory membership of an element to a set can be partial, i.e. an element belongs to a set with a certain grade (possibility) of membership. 3.3 Analytic Development of Reliability and Performance in Engineering Design 149 Fuzzy interpretations of data structures, p articularly during the initial stages of engineering design, are a very natural and intuitively plausible way to formulate and solve various design problems. Conventional (crisp) sets contain objects that satisfy precise properties required for membership. For example, the set of numbers H from 6 to 8 is crisp and can be defined as: H = {r ∈ R|6 ≤ r ≤ 8} Also, H is described by its membership (or characteristic) function (MF): m H : R →{0,1} defined as: m H (r)={16≤ r ≤ 8} = {0otherwise} Every real number r either is or is not in H.Sincem H maps all real numbersr ∈R onto the two points (0, 1), crisp sets correspond to two-valued logic: is or is not, on or off, black or white, 1 or 0, etc. In logic, values of m H are called truth values with reference to the question: ‘Is r in H?’ The answer is yes if, and only if m H (r)=1; otherwise, no. Consider the set F of real numbers that are close to 7. Since the property ‘close to 7’ is fuzzy, there is not a unique membership function for F. Rather, the decision must be made, based on the potential application and properties for F,whatm H should be. Properties that might seem plausible for F include: i) normality (i.e. MF(7)=1) ii) monotonicity (the closer r is to 7, the closer m H (r) is to 1, and conversely) iii) symmetry (numbers equally far left and right of 7 should have equal memberships). Given these intuitive constraints, functions that usefully represent F are m F1 , which is discrete (represented by a staircase graph), or the function m F1 , which is continu- ous but not smooth (represented by a triangle graph). One can easily construct a membership (or characteristic) function (MF) for F so that every number has some positive membership in F but numbers ‘far from 7’, such as 100, would not be expected to be included. One of the greatest differences between crisp and fuzzy sets is that the former always have unique MFs, whereas every fuzzy set may have an infinite number of MFs. This is both a weakness and a strength, in that uniqueness is sacrificed but with a gain in flexibility, enabling fuzzy models to be adjusted for maximum utility in a given situation. In conventional set theory, sets of real objects, such as the numbers in H,are equivalent to, and isomorphically described by, a unique membership function such as m H . However, there is no set theory with the equivalent of ‘real objects’ corre- spondingto m F . Fuzzy sets are always functions,from a ‘universe of objects’,say X, 150 3 Reliability and Performance in Engineering Design into [0,1]. The fuzzy set is the function m F that carries X into [0, 1]. Every function m: X →[0,1] is a fuzzy set by definition. While this is true in a formal mathematical sense, many functions that qualify on this ground cannot be suitably interpreted as realisations of a conceptual fuzzy set. In other words, functions that map X into the unit interval may be fuzzy sets, but become fuzzy sets when, and only when, they match some intuitively plausible semantic description of imprecise properties of the objects in X (Bezdek 1993). b) Formulation of Fuzzy Set Theory Let X be a space of objects and x be a generic element of X. A classical set A, A ⊆X, is defined as a co llection of elements or objects x ∈ X, such that each element (x) can either belong to the set A, or not. By defining a membership (or characteristic) function f or each element x in X, a classical set A can be represented by a set of ordered p airs (x,0), (x, 1), which indicates x /∈ A or x ∈ A respectively (Jang et al. 1997). Unlike conventional sets, a fuzzy set expresses the degree to which an element belongs to a set. Hence, the membership function o f a fuzzy set is allowed to h ave values between 0 and 1, which denote the degree of membership of an element in the given set. Obviously, the definition of a fuzzy set is a simple extension of the definition of a classical (crisp) set in which the characteristic function is permitted to have any values between 0 and 1. If the value of the membership function is restricted to either 0 o r 1, then A is reduced to a classical set. For clar ity, classical sets are referred to as ordinary sets, crisp sets, non-fuzzy sets, or just sets. Usually, X is referred to as the universe of discourse or, simply, the universe,and it may consist of discrete (ordered or non-ordered) objects or it can be a continuous space. The construction of a fuzzy set depends on two requirements: the identifi- cation of a suitable universe of discourse, and the specification of an appropriate membership function. In practice, when the universe of discourse X is a continuous space, it is partitioned into several fuzzy sets with MFs covering X in a more or less uniform manner. These fuzzy sets, which usually carry names that conform to adjectives appearing in daily linguistic usage, such as ‘large’, ‘medium’ or ‘small’, are called linguistic values or linguistic labels. Thus, the universe of discourse X is often called the linguistic variable. The specification of membership functions is subjective, which means that the membership functions specified for the same concept by different persons may vary considerably. This subjectivity comes from individual differences in perceiving or expressing abstract concepts, and has little to do with randomness. Therefore, the subjectivity and non-randomness of fuzzy sets is the primary difference between the study of fuzzy sets, and probability theory that deals with an objective view of random phenomena. 3.3 Analytic Development of Reliability and Performance in Engineering Design 151 Fuzzy Sets and Membership Functions If X is a collection of objects denoted generically by x, then a fuzzy set A in X is defined as a set of ordered p airs A = {(x, μ A(x))|x ∈ X},where μ A(x) is called the membership function (or MF, for short) for the fuzzy set A. The MF maps each ele- ment of X to a membership grade or membership value between 0 and 1 (included). More formally, a fuzzy set A in a universe of discourse U is characterised by the membership function μ A : U →[0,1] (3.93) The f unction associates, with each element x of U, a number μ A (x) in the inter- val [0,1]. This represents the grade of membership of x in the fuzzy set A.Forex- ample, the fuzzy term young might be defined by the fuzzy set given in Table 3.12 (Orchard 1998). Regarding Eq. (3.93), one can write: μ young (25)=1, μ young (30)=0.8, , μ young (50)=0 Grade of membership values constitute a possibility distribution of the term young. The table can be graphically represented as in Fig. 3.27. The possibility distribution of a fuzzy concept like somewhat young or very young can be obtained by applying arithmetic operations to the fuzzy set of the basic fuzzy term young,wherethemodifiers ‘ somewhat’and‘very’ are associated with specific mathematical functions. For example, the p ossibility values of each age in the fuzzy set representing the fuzzy concept somewhat young might be calculated by taking the square root of the corresponding possibility values in the fuzzy set of young, as illustrated in Fig. 3.28. These modifiers are commonly referred to as hedges. A modifier may be used to further enhance the ability to describe fuzzy con- cepts. Modifiers (very, slightly, etc.) used in phrases such as very hot or slightly cold change (modify) the shape of a fuzzy set in a way that suits the meaning of the word used. A typical set of predefined modifiers (Orchard 1998) that can be used to de- scribe fuzzy concepts in fuzzy terms, fuzzy rule patterns o r f uzzy facts is given in Table 3.13. Table 3.12 Fuzzy term young Age Grade of membership 25 1.0 30 0.8 35 0.6 40 0.4 45 0.2 50 0.0 152 3 Reliability and Performance in Engineering Design Possibility 1.0 0.0 µ young 10 20 30 40 50 60 70 80 Age Fig. 3.27 Possibility distribution of young Possibility 1.0 0.0 µ somewhat young 10 20 30 40 50 60 70 80 Age Fig. 3.28 Possibility distribution of somewhat young Table 3.13 Modifiers (hedges) and linguistic expressions Modifier name Modifier description Not 1−y Very y ∗∗ 2 Somewhat y ∗∗ 0.333 More-or-less y ∗∗ 0.5 Extremely y ∗∗ 3 Intensify (y ∗∗ 2) if y in [0,0.5] 1−2(1−y) ∗∗ 2ify in (0.5, 1] Plus y ∗∗ 1.25 Norm Normalises the fuzzy set so that the maximum value of the set is scaled 1.0(y = y ∗ 1.0/max-value) Slightly intensify (norm (plus A AND not very A)) = norm (y ∗∗ 1.25 AND 1−y ∗∗ 2) . Concept of Uncertainty in Engineering Design Analysis Introducing the concept of uncertainty in reliability assessment by utilising the tech- niques of FMEA and FMECA requires that some issues and. material used and the geometric d escription of the 146 3 Reliability and Performance in Engineering Design system(s) beingdesigned.Eventually,duringthe detail design phase ofthe engineer- ing design. Reliability and Performance in Engineering Design 149 Fuzzy interpretations of data structures, p articularly during the initial stages of engineering design, are a very natural and intuitively