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

However, typical of engineering systems, these fuzzy intervals may be divided by a landmark representing some critical quantity, with consequent uncertainty where the resulting point sho

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 stages 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 formalism refers to a specific physical quantity 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 << B A is much smaller than B

A − < B A is moderately smaller than B

A ∼< B A is slightly smaller than B

A == B A is exactly equal to B

A >∼ B A is slightly larger than B

A > − B A is moderately larger than B

A >> B A is 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 being 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 pressure 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 sign only, and differential equations are reinterpreted

as logical predicates The simulation involves finding values that satisfy these

con-straints (de Kleer et al 1984)

This work was further developed to represent the quantities by intervals and land-mark values (Kuipers 1986) Collectively, variables and landland-marks are described as

the quantities 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 ambiguities 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 landmark 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 ambiguity

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 converging on 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 points 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 In order, then, to consider the feasibility of qualiquali-tative reasoning in FMEA and FMECA without this resulting in ambiguity, it is essential to investigate further the concept of uncertainty in engineering design analysis

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 operates is defined The design engineer’s task is to establish a complete definition of the design and, in many 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 inner and outer

environments (Batill et al 2000) This is especially the case when several systems are integrated in a complex process with multiple (often conflicting) characteristics 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 through decisions 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 approximate 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 of this set of de-sign variables are central to, and will evolve with the dede-sign throughout the dede-sign process It is this quantitative description of the system, based 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 modelling techniques 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 of design variables will grow considerably Design variables typically are associated with the type of material used and the geometric description of the

system(s) being designed Eventually, during the detail design phase of the

engineer-ing design process, the designer will be required to specify (i.e quantify) the design variables representing the system 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 from 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 handbooks or 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

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 made 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 be-havioural knowledge, the propagation of system and equipment functions are also incomplete and uncertain In order to effectively capture uncertainty, a qual-itative approach is more appropriate to the available information than a quanti-tative one

In the following paragraphs, an overview is given of various concepts and theory for qualitatively modelling uncertainty in engineering design

3.3.2.4 Overview of Fuziness 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 not only to give such answers but also to describe their reality level, calculated from the uncertainty and imprecision of facts that are applicable The analysis should also be able to cope with

unreliable and incomplete information and with different expert opinions Many

commercial expert system tools or shells use different approaches to handle

uncer-tainty in knowledge or data, such as ceruncer-tainty 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 phases of the engineering design process

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 fuzziness are related but different concepts Fuzziness is a type of

deterministic uncertainty It describes the event class ambiguity Fuzziness measures

the degree to which an event occurs, not whether it does occur Probability 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 special case of fuzziness In essence, whenever the 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 of 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 generalisation of conventional set theory that was introduced

as a mathematical way to represent 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

Fuzzy interpretations of data structures, particularly 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) = {1 6 ≤ r ≤ 8}

= {0 otherwise}

Every real number r either is or is not in H Since m H maps all real numbers r ∈ 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, what m 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 mH (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-sponding to m Fuzzy sets are always functions, from a ‘universe of objects’, say X ,

into[0,1] The fuzzy set is the function mF 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 collection 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 for each element x in X , a classical set A can be represented by a set of ordered pairs (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 of a fuzzy set is allowed to have

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 or 1, then A is reduced to a classical set For clarity, 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 specifiidentifi-cation 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

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 pairs A = {(x,μA (x))|x ∈ X}, whereμA (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

The function associates, with each element x of U , a numberμA(x) in the

inter-val[0,1] This represents the grade of membership of x in the fuzzy set A For ex-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:

μyoung(25) = 1,μyoung(30) = 0.8, ,μyoung(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, where the modifiers ‘somewhat’ and ‘very’ are associated

with specific mathematical functions

For example, the possibility 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 or fuzzy facts is given in Table 3.13

Table 3.12 Fuzzy term young

Age Grade of membership

1.0

0.0

µyoung

Age

Fig 3.27 Possibility distribution of young

Possibility

1.0

0.0

µsomewhat young

Age

Fig 3.28 Possibility distribution of somewhat young

Table 3.13 Modifiers (hedges) and linguistic expressions

Modifier name Modifier description

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) ∗∗ 2 if y 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)