Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 14 pptx

3.3 Analytic Development of Reliability and Performance in Engineering Design 113
Determination of an o ptimum conceptual design is carried out as follows:
a) A performance parameter profile index (PPI)is calculated for each performance
parameter x
i
. This constitutes an analysis of the rows of the matrix, in which
PPI = n

n
∑
j= 1
1

c
ij

−1
(3.80)
where n is the number of design alternatives.
b) Similarly, a design alternative performance index (API) is calculated for each
design alternative y
j
. This constitutes an analysis of the columns of the matrix,
in which
API = m

m
∑
i=1
1


c
ij

−1
(3.81)
where m is the number of performance parameters.
c) An overall performance index (OPI) is then calculated as
OPI =
100
mn

m
∑
i=1
n
∑
j= 1
(PPI)(API)

(3.82)
where m is the number of performance parameters, n is the number of design
alternatives, and OPI lies in the range 0–100 and can thus be indicated as a per-
centage value.
d) Optimisation is then carried out iteratively to maximise the overall performance
index.
3.3.1.5 Labelled Interval Calculus
Interval calculus is a method for constraint propagation whereby, instead of des-
ignating single values, information about sets of values is propagated. Constraint
propagation of intervals is comprehensively dealt with by Moore (1979) and Davis
(1987). However, this standard notion of interval constraint propagation is not suf-

ficient for even simple design problems, which require expanding the interval con-
straint propagationconceptinto a new formalism termed “labelled interval calculus”
(Boettner et al. 1992).
Descriptions of conceptualas well as preliminary design represent sets of systems
or assemblies interacting under sets o f operating conditions. Descriptions of detail
designs represent sets of components functioning under sets of operating conditions.
The labelled interval calculus (LIC) formalises a system for reasoning about sets.
LIC defines a number o f operatives on intervals and equations, some of which can
be thought of as inverses to the usual notion of interval propagation by the question
‘what do the intervals mean?’ or, more precisely, ‘what kinds of relationships are
114 3 Reliability and Performance in Engineering Design
possible between a set of values, a variable,and a set of systems or components,each
subject to a set of operatingconditions?’. The usual notion of an intervalconstraintis
supplemented by the use of labels to indicate relationships between the interval and
a set of inferences in the design context. LIC is a fundamental step to understanding
fuzzy sets and possibility theory, which will be considered later in detail.
a) Constraint Labels
A constraint label describes how a variable is constrained with respect to a given
interval of values. The constraint label describes what is known about the values
that a variable of a system, assembly, or its components can have under a single set
of operating conditions.
There are four constraint labels: only, every, some and none. The best approach
to understanding the application of these four constraint labels is to give sample de-
scriptions of the values that a particular operating variable would have under a par-
ticular set of operating conditions, such as a simple example of a pump assembly
that operates under normal operating conditions at pressures ranging from 1,000 to
10,000kPa.
Only:
< only p 1000, 10000 > means that the pressure, under the specified operating
conditions, takes values only in the interval between 1,000 and 10,000kPa. Pressure

does not take any values outside this interval.
Every:
< every p 1000, 10000 > means that the pressure, under the specified operating
conditions, takes every value in the interval 1,000 to 10,000kPa. Pressure may or
may not take values outside the given interval.
Some:
< some p 1000, 10000 > means that the pressure, under the specified operating con-
ditions, takes at least one of the values in the interval 1,000 to 10,000kPa. Pressure
may or may not take values outside the given interval.
None:
< none p 1000, 10000 > means that the pressure, under the specified operating
conditions, never takes any of the values in the interval 1,000 to 10,000kPa.
b) Set Labels
A set label consolidates information about the variable values for the entire set of
systems or components under consideration. There are two set labels, all-parts and
some-part.
3.3 Analytic Development of Reliability and Performance in Engineering Design 115
All-parts:
All-parts means the constraint interval is true for every system or component in each
selectable subset of the set of systems under consideration. For example, in the case
of a series of pumps,
< All-parts only pressure 0, 10000 >
Every pump in the selected subset of the set of systems under consideration oper-
ates only under p ressures between 0 and 10,000kPa under the specified operating
conditions.
Some-part:
Some-part means the constraint interval is true for at least some system, assembly
or component in each selectable subset of the set of systems under consideration.
< Some-part every pressure 0, 10000 >
At least one pump in the selected subset of the set of systems under consideration

operates only under pressures between 0 and 10,000kPa under the specified operat-
ing conditions.
c) Labelled Interval Inferences
A method (labelled intervals) is defined for describing sets of systems or equipment
being considered for a design, as well as the operatives that can be applied to these
intervals. These labelled intervals and operatives can now be used to create inference
rules that draw conclusions about the sets of systems under consideration. There are
five types of inferences in the labelled interval calculus (Moore 1979):
• Abstraction rules
• Elimination conditions
• Redundancy conditions
• Translation rule
• Propagation rules
Based on the specifications and connections defined in the conceptual and pre-
liminary design phases, these five labelled interval inferences can be used to reach
certain conclusions about the integrity of engineering design.
Abstraction Rules
Abstraction rules are applied to labelled intervals to create subset labelled intervals
for selectable items. These subset descriptions can then be used to reason about the
design.
116 3 Reliability and Performance in Engineering Design
There are thr e e abstractio n rules:
Abstraction rule 1:
(only X
i
)(A
s,i
,S
i
) → (only x min

i
x
l,i
max
i
x
h,i
)(A∩
i
S
i
)
Abstraction rule 2:
(every X
i
)(A
s,i
,S
i
) → (every x max
i
x
l,i
min
i
x
h,i
)(A∩
i
S

i
)
Abstraction rule 3:
(some X
i
)(A
s,i
,S
i
) → (some x min
i
x
l,i
max
i
x
h,i
)(A∩
i
S
i
)
where
X = variable or ope rative interval
i = index over the subset
A = set of selectable items
A
s,i
= ith selectable sub set within set of selectable items
S

i
= set of states under which the ith subset operates
x = variable or operative
x
l,i
= lowest x in interval X of the ith selectable subset
min
i
x
l,i
= the minimum lowest value of x over all subsets i
max
i
x
l,i
= the maximum lowest value of x over all subsets i
x
h,i
= highest x in interval X of the ith selectable subset
min
i
x
h,i
= the minimum highest value of x over all subsets i
max
i
x
h,i
= the maximum highest value of x over all subsets i
∩

i
S
i
= intersection over all i subsets of the set of states.
Again, the best approach to understanding the application of labelled interval infer-
ences for describing sets of systems, assemblies or components being considered
for engineering design is to give sample descriptions of the labelled intervals and
their computations.
Description of Example
In the conceptual design of a typical engineering process, most sets of systems in-
clude a single process vessel that is served by a subset of three centrifugal pumps in
parallel. Any two of the pumps are continually operational while the third functions
as a standby unit. A basic design problem is the sizing and utilisation of the pumps
in order to determine an optimal solution set with respect to various different sets
of performance intervals for the pumps. The system therefore includes a subset of
three centrifugal pumps in parallel, any two of which are continually operational
while one is in reserve, with each pump having the following required pressure rat-
ings:
3.3 Analytic Development of Reliability and Performance in Engineering Design 117
Pressure ratings:
Pump Min. pressure Max. pressure
1 1,000kPa 10,000kPa
2 1,000kPa 10,000kPa
3 2,000kPa 15,000kPa
Labelled intervals:
X
1
= < all-parts every kPa 1000 10000 > (normal)
X
2

= < all-parts every kPa 1000 10000 > (normal)
X
3
= < all-parts every kPa 2000 15000 > (normal)
where
x
l,1
= 1,000
x
l,2
= 1,000
x
l,3
= 2,000
x
h,1
= 10,000
x
h,2
= 10,000
x
h,3
= 15,000
Computation: abstraction rule 2:
(every X
i
)(A
s,i
, S
i

) → (every x max
i
x
l,i
min
i
x
h,i
)(A∩
i
S
i
)
max
i
x
l,i
= 2,000
min
i
x
h,i
= 10,000
Subset interval:
< all-parts every kPa 2000 10000 > (normal)
Description:
Under normal conditions, all the pumps in the subset must be able to operate un-
der every value of the interval between 2,000 and 10,000 kPa. The subset interval
value must be contained within all of the selectab le items’ interval values.
Elimination Conditions

Elimination conditions determine those items that do not meet given specification s.
In order for these conditions to apply, at least one interval must have an all-parts la-
bel, and the state sets must intersect. Each specification is f ormatted such that there
are two labelled inter vals and a condition. One labelled interval describes a vari-
able for system requirements, while the other labelled interval describes the same
variable of a selectable subset or individual item in the subset.
There are three elimination conditions:
Elimination condition 1:
(only X
1
) and (only X
2
) and Not (X
1
∩X
2
)
118 3 Reliability and Performance in Engineering Design
Elimination condition 2:
(only X
1
) and (every X
2
) and Not (X
2
⊆ X
1
)
Elimination condition 3:
(only X

1
) and (some X
2
) and Not (X
1
∩X
2
)
Consider the example The system includes a subset of three centrifugal pumps in
parallel, any two of which are continually operational, with the following specifica-
tions requirement and subset interval:
Specifications:
System requirement: < all-parts only kPa 5000 10000 >
Labelled intervals:
Subset interval: < all-parts every kPa 2000 10000 >
where:
Pump1interval:< all-parts every kPa 1000 10000 >
Pump2interval:< all-parts every kPa 1000 10000 >
Pump3interval:< all-parts every kPa 2000 15000 >
Computation: elimination condition 2:
(only X
1
) and (every X
2
)andNot(X
2
⊆ X
1
)
Subset interval:

System requirement: X
1
=< kPa 5000 10000 >
Subset interval: X
2
=< kPa 2000 10000 >
Elimination result:
Condition: Not (X
2
⊆ X
1
) ⇒true
Description:
The elimination condition result is true in that the pressure interval of the subset
of pumps does not meet the system requirement, where
X
1
=< kPa 5000 10000 >
and the subset interval
X
2
=< kPa 2000 10000 >
A minimum pressure of the subset of pumps (kPa 2,000) cannot be less than the
minimum system requirement (kPa 5,000), prompting a review of the conceptual
design.
Redundancy Conditions
Redundancy conditions determine if a subset’s labelled interval (X
1
) is not signifi-
cant because another subset’s labelled interval (X

2
) is dominant.
3.3 Analytic Development of Reliability and Performance in Engineering Design 119
In order for the redundancy conditions to apply, the items set and the state set
of the labelled interval (X
1
) must be a subset of the items set and state set of the
labelled interval (X
2
). X
1
must have either an all-parts label or a some-parts label
that can be redundant with respect to X
2
, which in turn has an all-parts label.
Redundancy conditions do not apply to X
1
having an all-parts label while X
2
has
a some-parts label. Each redundancy condition is formatted so that there are two
subset labelled intervals and a condition.
There are five redundancy conditions:
Redundancy condition 1:
(every X
1
) and (every X
2
) and (X
1

⊆ X
2
)
Redundancy condition 2:
(some X
1
) and (every X
2
)and(X
1
∩X
2
)
Redundancy condition 3:
(only X
1
) and (only X
2
)and(X
2
⊆ X
1
)
Redundancy condition 4:
(some X
1
) and (only X
2
)and(X
2

⊆ X
1
)
Redundancy condition 5:
(some X
1
) and (some X
2
)and(X
2
⊆ X
1
)
Consider the example The system includes a subset of three centrifugal pumps in
parallel, any two of which are continually operational, with th e following specifica-
tions requirement and different subset configurations for the two operational units,
while the third functions as a standby unit:
Specifications:
System requirement: < all-parts only kPa 1000 10000 >
Pump 1 interval: < all-parts every kPa 1000 10000 >
Pump 2 interval: < all-parts every kPa 1000 10000 >
Pump 3 interval: < all-parts every kPa 2000 15000 >
Labelled intervals:
Subset configuration 1:
Subset1 interval: < all-parts every kPa 1000 10000 >
where:
Pump 1 interval: < all-parts every kPa 1000 10000 >
Pump 2 interval: < all-parts every kPa 1000 10000 >
120 3 Reliability and Performance in Engineering Design
Subset configuration 2:

Subset2 interval: < all-parts every kPa 2000 10000 >
where:
Pump1interval:< all-parts every kPa 1000 10000 >
Pump3interval:< all-parts every kPa 2000 15000 >
Subset configuration 3:
Subset3 interval: < all-parts every kPa 2000 10000 >
where:
Pump2interval:< all-parts every kPa 1000 10000 >
Pump3interval:< all-parts every kPa 2000 15000 >
Computation:
(every X
i
)(A
s,i
, S
i
) → (every x max
i
x
l,i
min
i
x
h,i
)(A∩
i
S
i
)
(every X

1
) and (every X
2
)and(X
1
⊆ X
2
)
For the three subset intervals:
1) Subset intervals:
Subset1 interval: X
1
=< kPa 1000 10000 >
Subset2 interval: X
2
=< kPa 2000 10000 >
Redundancy result:
Condition: (X
1
⊆ X
2
) ⇒false
Description:
The redundancy condition result is false in that the pressure interval of the pump
subset’s labelled interval (X
1
) is not a subset of the pump subset’s labelled inter-
val (X
2
).

2) Subset intervals:
Subset1 interval: X
1
=< kPa 1000 10000 >
Subset3 interval: X
2
=< kPa 2000 10000 >
Redundancy result:
Condition: (X
1
⊆ X
2
) ⇒false
Description:
The redundancy condition result is false in that the pressure interval of the pump
subset’s labelled interval (X
1
) is not a subset of the pump subset’s labelled inter-
val (X
2
).
3) Subset intervals:
Subset2 interval: X
1
=< kPa 2000 10000 >
Subset3 interval: X
2
=< kPa 2000 10000 >
Redundancy result:
Condition: (X

1
⊆ X
2
) ⇒true
Description:
The redundancy condition result is true in that the pressure interval of the pump
subset’s labelled interval (X
1
) is a subset of the pump subset’s labelled inter-
val (X
2
).
3.3 Analytic Development of Reliability and Performance in Engineering Design 121
Conclusion
Subset2 and/or subset3 combinations of pump 1 with pump 3 as well as pump 2
with pump 3 respectively are redundant in that pump 3 is redundant in the con-
figuration of the three centrifugal pumps in parallel.
Translation Rule
The translation rule generates new labelled intervals based on various interrelation-
ships among systems or subsets of systems (equipment). Some components have
variables that are directional. (Typically in the case of RPM, a motor produces
RPM-out while a pump accepts RPM-in.) When a component such as a motor has
a labelled interval that is being considered, the translation rule determines whether
it should be translated to a connected component such as a pump if the connected
components form a set with matching variables, and the labelled interval for the
motor is not redundant in the labelled interval for the pump.
Consider the example A system includes a subset with a motor, transmission and
pump where the motor and transmission have the following RPM ratings:
Component Min. RPM Max. RPM
Motor 750 1,500

Transmission 75 150
Labelled intervals:
Motor = < all-parts every rpm 750 1500 > (normal)
Transmission = < all-parts every rpm 75 150 > (normal)
Translation rule:
Pump = < all-parts every rpm 75 150 > (normal)
Propagation Rules
Propagation rules generate new labelled intervals based on previously processed
labelled intervals and a given relationship G, which is imp licit among a minimum
of three variables. Each rule is formatted so that there are two antecedent subset
labelled intervals, a given relationship G, and a resultant subset labelled interval.
The resultant labelled interval contains a constraint label and a labelled interval
calculus operative. The resultant labelled interval is determined by applying the
operative to the variables. If the application of the operative on the variables can
produce a labelled interval, a new labelled interval is propagated. If the application
of the operative on the variables cannot produce a labelled interval, the propagation
rule is not valid.
An item’s set and state set of the new labelled interval are the intersection of
the item’s set and state set of the two antecedent labelled intervals. If both of the
antecedent labelled intervals have an all-parts set label, the new labelled interval
122 3 Reliability and Performance in Engineering Design
will have an all-parts set label. If the two antecedent labelled intervals have any
other combination o f set labels (such as one with a some-part set label, and the
other with an all-parts set label; or both with a some-part set label), then the new
labelled interval will have a some-part set label (Davis 1987).
There are five propagation rules:
Propagation rule 1:
(only X) and (only Y)andG ⇒ (only Range (G, X,Y))
Propagation rule 2:
(every X) and (everyY)andG ⇒ (every Range (G, X, Y))

Propagation rule 3:
(every X) and (only Y) and state variable (z) or parameter (x)
and G ⇒ (every domain (G, X, Y))
Propagation rule 4:
(every X) and (only Y) and parameter (x)andG ⇒(only SuffPt (G, X, Y))
Propagation rule 5:
(every X) and (only Y)andG ⇒ (some SuffPt (G, X, Y))
Consider the example Determine whether the labelled interval of flow for dy-
namic hydraulic displacement pumps meets the system specifications requirement
where the pumps run at revolutions in the interval of 75 to 150RPM, and the pumps
have a displacement capability in the interval 0.5×10
−3
to 6 ×10
−3
cubic metre
per r evolution. Displacement is the volume of fluid that moves through a hydraulic
line per revolution of the pump impellor, and RPM is the revolution speed of the
pump. The flow is the rate at which fluid moves through the lines in cubic metres
per minute or per hour.
Specifications:
System requirement: < all-parts only flow 1.50 60 > m
3
/h
Given relationship:
Flow (m
3
/h) = (Displacement × RPM) ×C
where C is the pump constant based on specific pump characteristics.
Labelled intervals:
Displacement (

η
) = < all-parts only
η
0.5×10
−3
6×10
−3
>
RPM (
ω
) = < all-parts only
ω
75 150 >