Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 13 doc

3.2 Theoretical Overview of Reliability and Performance in Engineering Design 103
α
0
= the total number of items put on test at time t = 0,
α
S
= the number of items surviving at time t or at t + Δt.
Substituting Eq. (3.64) into Eq. (3.63) leads to
R
i
(t)=1−
t

0
f
i
(t)dt . (3.65)
A more common notation for the ith component reliability is expressed in terms of
the mathematicalconstant e. Themathematical constant e is the uniquereal number,
such that the value of the derivative of f(x)=e x at the point x = 0 is exactly 1.
The function so d efined is called the exponential function. Thus, the alternative,
commonly used expression for R
i
(t) is
R
i
(t)=e
−

t
0

λ
i
(t) dt
, (3.66)
where
λ
i
(t) is the ith component hazard rate or instantaneous failure rate.
In this case, component failure time can follow any statistical distribution func-
tion of which the hazard r ate is known. The expression R
i
(t) is r e duced to
R
i
(t)=1−F
i
(t)
R
i
(t)=e
−
λ
i
t
. (3.67)
A redundant configuration or single component MTBF is defined by
MTBF =
∞

0

R(t)dt . (3.68)
Thus, substituting Eq. (3.67) into Eq. (3. 66), and in tegr ating the results in the series
gives the model for MTBF, which in effect is the sum of the inverse values of the
component hazard rates, or instantaneous failure rates o f all the components in the
series
MTBF =

n
∑
i=1
λ
i

−1
(3.69)
MTBF = sum of inverse values of component hazard rates
= instantaneous failure rates of all the components.
b) Parallel Network
This type of redundancy can be used to improve system and equipment reliabil-
ity. The redundant system or equipment will fail only if all of its components fail.
To develop this mathematical model for application in reliability evaluation,itis
104 3 Reliability and Performance in Engineering Design
assumed that all units of the system are active and load sharing, and units are sta-
tistically independent. The unreliability, F
P
(t), at time t of a parallel structu re with
non-identical components is
F
P
(t)=

k
∏
i=1
F
i
(t) (3.70)
F
i
(t)=ith component unreliability (failure probability).
Since R
P
(t)+F
P
(t)=1, utilising Eq. (3.70) the parallel structure reliability, R
P
(t),
becomes
R
P
(t)=1−
k
∏
i=1
F
i
(t) . (3.71)
Similarly,as was donefor the series network componentswith constant failure rates,
substituting for F
i
(t) in Eq. (3.71) we get

R
P
(t)=1−
k
∏
i=1

1− e
−
λ
i
t

. (3.72)
In order to obtain the series MTBF, substitute Eq. (3.69) for identical components
and integrate as follows
MTBF =
∞

0

1−
k
∑
j= 0
(n
j
)(−1)
j
e

−
λ
j
t

dt
MTBF =
1
λ
+
1
2
λ
+
1
3
λ
+ +
1
k
λ
(3.73)
λ
= the component hazard or instantaneous failure rate.
c) A k-out-of-m Unit Network
This type of redundancy is used when a certain number k of components in an ac-
tive parallel redundant system orassembly must work forthe system’s or assembly’s
success. The binomial distribution, system or assembly reliability of the indepen-
dent and identical components at time t is R
k/m

(t),whereR(t) is the component
reliability
R
k/m
(t)=
m
∑
i=k
t
(m)[R(t)]
i
[1−R(t)]
k−i
(3.74)
m = the total number of system/assembly components
k = the number of components required for system/assembly success
at time t.
3.2 Theoretical Overview of Reliability and Performance in Engineering Design 105
Special cases of the k-out-of-m unit system are:
k = 1: = parallel network
k = m: = series network.
For exponentially distributed failure times (constant failure rate) o f a component,
substituting in Eq. (3.74) for k = 2andm = 4, the equation becomes
R
2/4
(t)=3e
−4
λ
t
−8e

−3
λ
t
+ 6e
−2
λ
t
. (3.75)
d) Standby Redundant Systems
R
S
(t)=
K
∑
i=0
⎡
⎣
t

0
λ
(t)dt
⎤
⎦
i
e
−

t
0

λ
(t) dt
(i!)
−1
. (3.76)
In this case (Eq. 3.76), one component is functioning, and K components are on
standby, or are not active. To develop a system/assembly reliability model, the com-
ponents must be identical and independent, and the standby components as new.
The general components hazard rate,
λ
, is assumed.
3.2.3.5 Reliability Evaluation of Three-State Device Networks
A three-state device (component) has one operational and two failure states. De-
vices such as a fluid flow valve and an electronic diode are examples of a three-
state device. These devices have failure modes that can be described as failure in
the closed or open states. Such a device can have the following functional states
(Dhillon 1983):
State 1 = Operational
State 2 = Failed in the closed state
State 3 = Failed in the o pen state
a) Parallel Networks
A parallel network composed of active independent three-state componentswill fail
only if all the components fail in the open mode, or at least one of the devices must
fail in the closed m ode. The network (with non-identical devices) time-dependent
reliability, R
P
(t),is
R
P
(t)=

k
∏
i=1
[1−F
C
i
(t)] −
k
∏
i=1
F
O
i
(t) , (3.77)
106 3 Reliability and Performance in Engineering Design
where:
t = time
k = the number of three-state devices in parallel
F
C
i
(t)= the closed mode probability of device i at time t
F
O
i
(t)= the open mode probability of device i at time t
b) Series Networks
A series network is the reverse of the parallel network. A series system will fail only
if all of its independent elements fail in a closed mode or any one of the components
fails in open mode. Th us, because of duality, the time-dependen t reliability of the

series network with non-identical and independent devices is the difference of the
summations of the respective values for the open mode probability,[1−F
O
i
(t)], and
the closed mode probability,[F
C
i
(t)], of device i at time t.
The series network with non-identical and independent devices time-dependent
reliability, R
S
(t),is
R
S
(t)=
k
∏
i=1
[1−F
O
i
(t)] −
k
∏
i=1
F
C
i
(t) , (3.78)

where:
t = time
k = the number of devices in the series configuration
F
C
i
(t)= the closed mode probability of device i at time t
F
O
i
(t)= the open mode probability of device i at time t
Closing comments to theoretical overview
It was stated earlier, and must be iterated here, that these techniques do not represent
the total spectrum of reliability calculations, and have been considered as the most
applicable for their application in determining the integrity of engineering design
during the conceptual, preliminary and detail design phases of the engineering de-
sign process, based on an extensive study of the available literature. Furthermore, the
techniques h ave been grouped according to significant differences in the approaches
to the determination of reliability of systems, compared to that of assemblies or of
components. This supports the premise that:
• predictions of the reliability of systems are based on prognosis of systems perfor-
mance under conditions subject to failure modes (reliability prediction);
• assessments of the reliability of equipment are based upon inferences of failure
according to various statistical failure distributions (reliability assessment); and
• evaluations of the reliability of components are based upon known values of fail-
ure rates (reliability evaluation).
3.3 Analytic Development of Reliability and Performance in Engineering Design 107
3.3 Analytic Development of Reliability and Performance
in Engineering Design
Some of the techniques identified fo r reliability prediction, assessment and evalua-

tion, in the conceptual, preliminary and detail design phases respectively, have been
considered for further analytic development.This has been done on the basis of their
transformational capabilities in developing intelligent computer automated method-
ology. The techniques should be suitable for application in artificial intelligence-
based modelling, i.e. AIB modelling in which knowledge-based expert systems
within a blackboardmodel can be applied in determiningthe integrityof engineering
design. The AIB model should be suited to applied concurrent engineering design in
an online and integrated collaborative engineering d esign environment in which au-
tomated continual design reviews are conducted throughout the engineering design
process by remotely located design groups communicating via the internet.
Engineering designs are usually composed of highly integrated, tightly coupled
systems with complex interactions, essential to the functional performance of the
design. Therefore, concurrent, rather than sequential considerations of specific re-
quirements are essential, such as meeting the design criteria together with design
integrity constraints. The traditional approach in industry for designing engineered
installations has been the implementation of a sequential consideration of require-
ments for process, thermal, power, manufacturing,installation and/or structural con-
straints. In recent years, concurrent engineering design has become a widely ac-
cepted concept, particularly as a pre ferred altern ative to the sequential engineerin g
design process. Concurrent engineering design in the context of design integrity is
a systematic ap proach to integratin g the various continual design reviews within the
engineering design process, such as reliability prediction, assessment, and evalua-
tion throughout the preliminary, schematic, and detail design phases respectively.
The objective of concurrent engineering design with respect to design integrity is
to assure a reliable design throughout the engineering design process. Parallelism
is the prime concept in concurrent engineering design, and design integrity (i.e. de-
signing for reliability) becomes the central issue. Integrated collaborative engineer-
ing design implies information sharing and decision coordinationfor conductingthe
continual design reviews.
3.3.1 Analytic Development of Reliability and Performance

Prediction in Conceptual Design
Techniques for reliability and performance prediction in determining the integrity
of engineering design during the conceptual design phase include system reliability
modelling based on:
i. System performance measures
ii. Determination of the most reliable design
108 3 Reliability and Performance in Engineering Design
iii. Conceptual design optimisation and
iv. Comparison of conceptual designs
v. Labelled interval calculus and
vi. Labelled interval calculus in designin g for reliability
3.3.1.1 System Performance Measures
For each process system, there is a set o f performance measures that require particu-
lar attention in design—for example, temperature range, pressure rating, output and
flow rate. Some measures such as pressure and temperature rating may be common
for different items of equipment inherent to each process system. Some measures
may apply only to one system. The performance measures of each system can be
described in matrix form in a parameter profile matrix (Thompson et al. 1998), as
showninFig.3.22where:
i = number of performance measure parameters
j = number of process systems
x = a data point that measures the performance of a system with respect to
a par ticular parameter.
It is not meaningful to use actual performance—for example, an operating
temperature—as the value of x
ij
. Rather, it is the proximity of the actual perfor-
mance to the limit of process capability of the system that is useful.
In engineering design review, the proximity of performance to a limit closely
relates to a measure of the safety margin. In the case of process enhancement, the

proximity to a limit may even indicate an inhibitor to proposed changes. For a pro-
cess system, a non-dimensionalnumerical value of x
ij
may be obtained by determin-
ing the limits of capability ,suchasC
max
and C
min
, with respect to each performance
parameter, and specifying the nominal point or range at which the system’s perfor-
mance parameter is required to operate.
The limits may be r epresented diagrammatically as shown in Figs. 3.23, 3.24
and 3.25, where an example of two performance limits, of one upper performance
limit, and of one lower performance limit is given respectively (Thompson et al.
1998).
The data point x
ij
that is entered into the performance of systems with two p er-
formance limits is the lower value of A and B (0 < score < 10), which is the closest
Process systems
Performance x
11
x
12
x
13
x
14
x
1i

parameters x
21
x
22
x
23
x
24
x
2i
x
31
x
32
x
33
x
34
x
3i
x
j1
x
j2
x
j3
x
j4
x
ji

Fig. 3.22 Parameter profile matrix
3.3 Analytic Development of Reliability and Performance in Engineering Design 109
Fig. 3.23 Determination of a data point: two limits
Fig. 3.24 Determination of a data point: one upper limit
the n ominal design co ndition does approach a limit. The value of x
ij
always lies in
the range 0–10. Ideally, when design condition is a single point at the mid-range,
then the data point is 10.
110 3 Reliability and Performance in Engineering Design
Fig. 3.25 Determination of a data point: one lower limit
It is obvious that this process of data point determination can be generated
quickly by computer modelling with inputs from process system performance mea-
sures and ranges of capability. If there is one operating limit only, then the data
point is obtained as shown in Figs. 3.24 and 3.25, where the upper or lower limits
respectively are known.
Therefore, a set of data points can be obtained for each system with respect to
the performance parameters that are relevant to that system. Furthermore, a method
can be adopted to allow designing for reliability to be quantified, which can lead to
optimisation of design reliability.
Figures 3.23, 3.24 and 3.2 5 illustrate how a data point can be generated to mea-
sure performance with respect to the best and the worst limits of pe rformance.
3.3.1.2 Determination of the Most Reliable Design
in the Conceptual Design Phase
Reliability prediction through system reliability modelling based on system perfor-
mance may be carried out by the following method (Thompson et al. 1999):
a) Identify the criteria against which the process design is measured.
b) Determine the maximum and minimum acceptable limits of performance for
each criterion.
c) Calculate a set of measurement data points of x

ij
for each criterion according to
the algorithms indicated in Figs. 3.23, 3.24 and 3.25.
3.3 Analytic Development of Reliability and Performance in Engineering Design 111
d) A design proposal that has good reliability will exhibit uniformly high scores
of the data points x
ij
. Any low data point represents system performance that is
close to an unacceptable limit, indicating a low safety margin.
e) The conceptual design may then be reviewed and revised in an iterative manner
to improve low x
ij
scores.
When a uniformly high set of scores has been obtained, then the design, or alter-
native design that is most reliable, will conform to the equal strength principle,also
referred to as unity, in which there are no ‘weak links’ (Pahl et al. 1996).
3.3.1.3 Comparison of Conceptual Designs
If it is required to compare two or more conceptual designs, then an overall rating
of reliability may be obtained to compare these designs. An overall reliability may
be determined by calculating a systems performance index (SP) as follows
SP = N

N
∑
i=1
1

d
i


−1
(3.79)
where
N = the sum of the p erformances considered
d
i
= the scores of the performances considered.
The overall SP score lies in the range from 0 to 10. The inverse method of combina-
tion o f scores readily identifies low safety margins, unlike normal averaging through
addition where almost no safety margin with respect to one criterion may be com-
pensated for by high safety margins elsewhere—which is unacceptable. Alternative
designs can therefore be compared with respect to reliability, by comparing their
SP scores; the highest score is the most reliable. In a proposed method for using
this overall rating approach (Liu et al. 1996), caution is required because simply
choosing the highest score may not be the best solution. This requires that each de-
sign should always be reviewed to see whether weaknesses can be improved upon,
which tends to defeat the purpose of the method. Although other factors such as
costs may be the final selection criterion for conceptual or preliminary design pro-
posals with similar overall scores (which oft is the case), the objective is to achieve
a design solution that is the most reliable from the viewpoint of meeting the re-
quired performance criteria. This shortcoming in the overall rating approach may
be avoided by supplementing performance measures obtained from mathematical
models in the form of mathematical algorithms of process design integrity for the
values of x
ij
, rather than the ‘direct’ performance parameters such as temperature
range, pressure rating, output or flow rate.
The performance measures obtained fro m these mathematical models consider
the prediction, assessment or evaluation of parameters particular to each specific
stage of the design process, whether it is conceptual design, preliminary design or

detail design respectively.
112 3 Reliability and Performance in Engineering Design
The approach defines performancemeasures that, when met, achievean optimum
design with regard to overall integrity. It seeks to maximise the integrity of design
by ensuring that the criteria of reliability, availability, maintainability and safety are
concurrently being met. The choice of limits of performance for such an approach is
generally made with respect to the consequences and effects of failure, and reliabil-
ity expectations based on the propagation of single maximum and minimum values
of acceptable performance for each criterion. If the consequences and/or effects of
failure are h igh, then limits of acceptable performance with high safety margins that
are well clear of failur e criteria are chosen. Similarly, if failure criteria are imprecise,
then high safety margins are adopted.
These considerationshave been further expanded to represent sets of systems that
function under sets of failures and performance intervals, applying labelled interval
calculus (Boettner et al. 1992).
The most significant advantage of this expanded method is that, besides not hav-
ing to rely on the propagation of single estimated values of failure data, it also does
not have to rely on the determination of single values of maximum and minimum
acceptable limits of performance for each criterion. Instead, constraint propaga-
tion of intervals about sets of performance values is applied. As these intervals are
defined, it is possible to compute a multi-objective optimisation of performance val-
ues, in order to d etermine optimal solution sets for different sets of performance
intervals.
3.3.1.4 Conceptual Design Optimisation
The process described attempts to improvereliability continually towards an optimal
result (Thompson et al. 1999). If the design problem can be modelled so that it is
possible to compute all the x
ij
scores, then it is possible to optimise mathematically
in order to maximise the SP function, as a result of which the x

ij
scores will achieve
a uniformly high score. Typically in engineering design, several conceptual design
alternatives need to be optimised for different design criteria or constraints.
To deal with multiple design alternatives, the parameter profile matrix,inwhich
the scores for each system’s performance measure of x
ij
is calculated, needs to be
modified. Instead of a one-variable matrix, in which the scores x
ij
are listed, the
analysis is completed for each specific criterion y
j
. Thus, a two-variable matrix of
c
ij
is constructed, as shown in Fig. 3.26 (Liu et al. 1996).
Design alternatives y
1
y
2
y
3
y
4
y
n
Performance x
1
c

11
c
12
c
13
c
14
c
1n
parameters x
2
c
21
c
22
c
23
c
24
c
2n
x
3
c
31
c
32
c
33
c

34
c
3n
x
m
c
m1
c
m2
c
m3
c
m4
c
mn
Fig. 3.26 Two-variable parameter profile matrix