152
| Date post: | 27-Nov-2014 |
| Category: |
Documents |
| Upload: | ashok-kumar |
| View: | 43 times |
| Download: | 0 times |
Ten most important product attributes
Attribute Average score
Performance 9.5
Lasts a long time (reliability) 9.0
Service 8.9
Easily repaired (maintainability) 8.8
Warranty 8.4
Easy to use 8.3
Appearance 7.7
Brand name 6.3
Packaging/display 5.8
Latest model 5.4
Definition of Reliability
The Probability of an item ( component/system) to perform adequately for a specified period of time. under stated conditions
Important Concerns• Probability
• Adequate Performance
• Time
• Conditions of use
Failure characteristics
Mortality curve
The bath tub curve : Table Characterized by Caused by Reduced by
Burn-in DFR Manufacturing defects: Burn-in testing Welding flaws, cracks, Screening Defective parts , Quality control Poor quality control, Acceptance testing Contamination, poor Workman ship Useful life CFR Environment Redundancy Random loads Excess strength “Acts of God” Chance events
Wear-out IFR Fatigue Derating Corrosion Preventive maintenance Aging Parts replacement Friction Technology Cyclical Loading
generation years Technology Performance
Switching device Storage device Switching time
MTBF
First 1949-55 Vacuum tubes Acoustic delay lines and later magnetic drums.
1Kbyte memory
0.1 to 1ms 0.5 to 1h
Second 1956-65 Transistors Magnetic core main memory, tapes and disks
as peripheral memory 100Kbyte main memory
1 to 10µs About 10hrs
Third 1966-75 Integrated circuits (IC) High speed magnetic cores. Large disks (100 MB). 1 MB
main memory
0.1 to 1 µs About 100 hours
Fourth 1975-91 Large sclae integrated circuits. Microprocessor
(LSI)
Semi conductor memory. Winchester disk. 10MB main memory. 1000 MB
disks
10 to 100ns About 1000
hourts
Fifth 1991- VLSI, Associative memories. Optical disks. 10GB of
distribute storage
0.1ns speed measure in
gig lips
10000 hours
Comparison of performance of successive generations of digital computers
System Modeling
– Series model
– Parallel model
– Series parallel model
– Parallel series model
– K out of M models– Non Series Parallel Models
ASSUMPTIONS
1 Each unit and the system has two states i.e. either good or failed
2 Unit states are s-independent
3 Sensing and switching of failed unit out of the system is perfect
4 Unit reliabilities are not necessary equal and reliability of each unit is known
5 The system is good if and only if at least k of its n units are good.
6 Unit of the system are numbered from 1 to n.
Series Model :Automobile with four Wheels
Characteristics:
• All components must operate successfully for the system success.
• System reliability ,Rs=
• MTTF=
0)( dttRs
n
ii
1
1
n
ii
1
Mean Time to Failure-MTTF
0 0
0 0 0
( ) ( )
( ) ( ) ( )
1
MTTF tf t dt tR t dt
tR t R t dt R t dt
Parallel model: Two Boilers But needed
One for operation
Characteristics:
Rs =
=
)1(11
m
ii
0
)( dttRsMTTF
m
ii
q1
1
m
i
i1
)1(
Series- Parallel model
Characteristics:
Rs =
=
=
=
)}1(1{1 1
n
i jij
mi
})1(1{1
min
ii
})1(1{1
mn
ii
nmP ])1(1[
smallqnqq mnm ,1)1(
dteMTTF nmt ])1(1[0
Parallel – Series model
characteristics:
Rs = 1- }1{1 1
m
i jij
ni
}1{11 1
m
i
n
jij
}1{11
m
i
n
i
mn}1{1
dteMTTF mtn ))1(1(0
k out of m models
• Probability of exactly k successes out of m is
kmk
k
pppmk m )1()(),;(
imim
ki i
ppRs m
)1()(
pqqqRs imim
kmi im
1,)1()(1
1
K-out-of-n: G(F) systems
k-out-of-n: G(F) systems• Introduced by Birnbaum [42] in 1961
• Most general model in the studies of Series –Parallel System
DEFINITION
A k-out-of-n :G(F) system is a system of n components in which at least k components must function (fail) for the system to function (fail).
K-out-of –n :G system is exactly equivalent to (n-k+1)-out-of-n:F system
SPECIAL CASES
for
k= 0 perfectly reliable system
k=1 parallel system
k= (n+1)/2 NMR system (voting)
k= n-1 Fail safe system
k= n Series system
1<k<n Partially redundant system
Examples:
Reactor Protection System
Sensor System
Alarm generating System
A bus structured microprocessor computer system
consisting N processors and sharing M memory unit
via Z buses
A piece of stranded wire with n strands out of which
at least k - strands are necessary to pass the
required current
Voting networks
Automobile air bag restraining system
ASSUMPTIONS
1 Each unit and the system has two states i.e. either good or failed
2 Unit states are s-independent
3 Sensing and switching of failed unit out of the system is perfect
4 Unit reliabilities are not necessary equal and reliability of each unit is known
5 The system is good if and only if at least k of its n units are good.
6 Unit of the system are numbered from 1 to n.
Reliability Evaluation of k-out-n :G System
For identical unit
…(1)
Where q= 1-p
(1) Utilizes binomial expansion , B(k, n, p) = c( n k) pk (1-p)n-k
Binomial expansion cannot be used to evaluate reliability of k-out-of-n :G system with unidentical components.
inin
ki
ppincnkR
1,,
inin
ki
ppinc
1,
Reliability Evaluation of k-out-of-n:G System with Unidentical Units
Basic Difficulties
1. To determine the Sets of minimum components whose success will ensure success of the system.
Total number of minimum success terms = c(n,k).
Example for 4-out-of-6 :G system , number of minimum success terms are c(6,4)=15
2. To obtain reliability expression
Different Basic Method /Technique Used
TECHNIQUE 1 Let number of minimum Success term used be,
c(k,n) = m
System success function
Where each Ti is a min success term & m = c(n,
k)
mm
m
i kjikji
jijii
TTTTP
TTTPTTPTPnkR
...1...
...,
3211
1
m
iiTSSSF
1
,
For (n-k+1) –out-of- n:F system :
Number of minimum failed terms c(n-k+1,n)=m
R(k,n)= 1 - U(k,n)
Where
c(k,n) and c(n-k+1,n) have been termed as min terms
Total no of terms in reliability expression =2m –1
Technique is cumbersome and time consuming
Both canceling and non canceling terms are generated.
mm
m
i kjikji
jijii
CCCCP
TTTPCCPCPnkU
....1......
...,
3211
1
TECHNIQUE - 2•S1.
•S2
•S3
Efficiency of the technique depends upon how Sdisj expression in S2 is obtained.
m
iiTSSSF
1
,
dm
jjdisj TS
1
dm
jjrdisjr TPSP
1
dmrrr TPTPTP ...21
TECHNIQUE - 3
WhereReliability expression is obtained by changing
&
n
iidisj TSSSF
1
,
n
jir
n
iir TPTPnKR
11
,
tSri XXXT ......
tt
sS
rr
pX
qX
pX
TECHNIQUE - 4System
(1) is in sum of disjoint products.
R(k,n) is obtained by recursive use of (1) for different values of k.
)...(1,11,, InkSnkSXnkS i
)...(,11,1, IIknRqknRpnkR ii
Classification of k-out-of-n: G Systems
Technique 1 Inclusion-Exclusion (IE) Method.
Technique 2 Disjoint Product (DP) Method
Technique 3 State Enumeration (SE)
Method
Technique 4 Decomposition Technique
INFERENCES1. Number of terms of the reliability expression, number of multiplications / divisions and number of additions / subtractions involved in these terms can not be the sole criterian of computational complexity.
2. Most suitable algorithm can be chosen only if the computational costs and the specific requirements of the application in case of symbolical evaluation have been considered.
Illustration: Suppose the component reliabilities are given by symbolic expressions. Aim is to determine sensitivity or MTTF. Then considering only the number of multiplications in the comparison of different algorithms does not give a fair picture of computational costs.
Application (say MTTF to be computed symbolically) depends upon the symbolic reliability expression.
Consider three reliability expressions for 2 – out – of – 3: G system
(i) p1 p2 + (p1 + (q1 – p1) p2 – p1 p2) p3 + p1 p2 p3 [BH – I]
(ii) p1 p2 q3 + p1 q2 p3 + q1 p2 p3 + p1 p2 p3 (state enumeration)
(iii) p1 p2 + p1 p3 + p2 p3 – 2 p1 p2 p3
The most suitable is expression (iii) 3. For a true comparison, jumps to sub expressions (recursive algorithms) have to be accounted for.4. Temporal complexity in most of the methods depends upon the
version (reliability or unreliability) of the algorithm. AR [47] has same complexity for reliability and unreliability versions. 5. Inconsistency in the computational complexity of the algorithms needs to be investigated and resolved as computational complexity in most of these have been compared on the basis of multiplications / divisions, and additions / subtractions of the terms involved in reliability expressions.
The statement: A problem that has taken more than its share in the literature [54], is true with regards to plethora of information available in terms of algorithms but computational complexity needs to be investigated further.
6. Controversy with regard to computational existed in four
algorithms mainly these are JG [49], RSPK [51], SP [55] and
BH [46]. BH [46] supersedes [48, 53, 54] JG [49], RSPK [51]
and SP [55] & probably optimal [48]. BH-1 has better worst
case bound and RSPK-1 has the worst case computational
bound. JG’s [49] worst case bounds are in between BH –I and
RSPK-I.
7. State Enumeration is a worst case of disjointing products
(DPs) algorithms. BH [46] employs state enumeration
approach and yields optimal solution temporally.
8. Method of Behr et [57] utilizes IE law for obtaining terms and
factoring for signed domination [78].
& d (k, n: G) =
for n = k d (k, k: G) =
111
k
kn n
0111
kn
Classification of k-out-of-n: G Systems.
Recursive Non-Recursive
StateEnumeratio
n
InclusionExclusion
DisjointProduct
Decomposition
AR [47]RSPK [51]
JG [49]BCP [57]
Haskin [56]
BH [46]Locks [61]MG [50]BM [43]SP [55]
Huang [59]
Rissel [54]Pham et al [53]Rushdi [48]
Comparison of Complexity Related issues of algorithms.
Capacity-k-out-of-n System Reliability Evaluation
• A new measure of k-out-of-n: G (F) System.
• Unlike recursive algorithm advanced by Wu et al [58], the
proposed method is a non-recursive technique.
DEFINITION:
• A capacity-k-out-of-n: G (F) System is a System of n
components, each with its own positive integer transmission
capacity such that the system is good (failed) if and only if
the total capacity of good (failed) components is at least k
[58].
• The reliability of capacity-k-out-of-n: G System is the
complement of the unreliability of capacity-(w – k + 1) – out –
of - n: F System where, w is the total capacity of the system.
APPLICATIONS:
Supply systems comprising components
with fixed ratings for their capacity, flow,
throughput. The system is said to be successful
when it is able to transmit a specified minimum
flow.
ASSUMPTIONS:
1. Each component has a fixed transmission
capacity.
2. The system is good iff the total capacity of
good components is at least k.
METHOD The method determines all the minimal success terms
satisfying the flow constraint. The procedure is given as under:
S1 Capacity-k of the system to be transmitted is given. Let it be wmin. The maximum capacity wmax of the system is
obtained by adding the capacity of all the n components of the system. Components are numbered in a sequence according to decreasing order of their capacity & let it be represented by set A. Also define a set B for minimal success terms to start with it is a null set.
S2 If the maximum transmission capacity of the system is greater than or equal to the minimum transmission capacity of the network, then solution of the problem exists.
S3 If the capacity of first element is greater than or
equal to wmin, first element of the set A is
transferred to set B. The procedure is repeated for
all the elements of set A.
The cardinality of the set A reduces and is equal to
the difference between the total number of
components present initially and the number of
components transferred to set B. Let the set A with its
reduced cardinality be termed as AR (1).
S4 The above process from S2 to S3 is repeated
considering all the combinations of two components at a
time for the set AR (1) & is explained as given below:
If the maximum capacity of set AR (1) is greater than or equal to wmin, then obtain the sum of capacity of first two components. If it is greater than or equal to wmin then product (intersection) of these two components is copied to set B. The procedure is repeated for all the combinations of two components such that first component is included in each combination. When all such combinations have been exhausted the cardinality of set AR (1) is reduced to AR (2) by deleting first element of AR (1). The procedure adopted for AR (1) is now repeated with set AR (2), AR (3) and so on.
The process from S2 to S4 is repeated for all the combinations of three components, four components & so on. The elements of set B are the possible minimal success terms satisfying the flow constraint. A disjoint sum-of-products is obtained using any of the existing disjointing technique. The reliability expression in obtained by changing logical variables of the terms present in disjoint expression into corresponding probability variables. The algorithm developed for the capacity-k-out-of-n: G (F) system is given as under:
ALGORITHM S1: Initialization
Input wi, i =1 to n in decreasing order, , wmin= 5; B =
. Arrange the components i = 1, in decreasing order with regard
to their capacity.
i.e. A = [X1 X2 X3 X4 X5]
s.t. w1 = 5; w2 = 4; w3 = 3; w4 = 3; w5 = 2;let i = 1, w = [5 4 3 3 2]
S2: If wmax wmin, Go to next step (S3).S3: Otherwise Go to S6.
(i) if (wi - wmin)
(ii) {B} = {B} {Xi}
(iii) {A} = {A} – {Xi}
(iv) wmax = wmax - wi
(v) {w} = {w} - wi
Increment i, & go to S2.
n
iiww
1max
S4: A (I) = W (I) = X (J)For I = i, j = i +1 to n
R = wmin - wi
R1 = Xj+1 – R
If (Xj – R)
B = B Xi Xj
Increment j (until j > n end)
[A] = [A] – [Xi]
wmax = wmax - wi
[W] = [W] – [wi]Increment i, & go to S2.W (I) = X (J) = Y (K)
For I = i, I = i + 1 to n – 1, K = i + 2 to n
Do begin R2 = R – Xj
S5: If (Xj+1 – R2)
B = B Xi Xj Xk
Increment k and
A = [A] - Xi
wmax = wmax – wi
[W] = [W]– [Wi]increment i.Repeat steps for four variables at a time and so on.
S6: B = Xi Xi Xj Xi Xj Xk …….
Obtain Bdisj .
S7: Reliability expression is obtained by changing logical variables into corresponding probability variables, pi (qi).
ILLUSTRATIONS Example A system has five components. The capacity of each component is shown in Table 1.
Reliability of capacity-5-out-of-5: G system is to be determined.Solution:
S1 Initialization Input-Set A = {X1, X2, X3, X4, X5}
Input wi, i = 1 to 5. Arrange capacity of components in decreasing order, represented by a se W.
Let W = { 5 4 3 3 2}; B =
k = wmin = 5 (given) Let i = 1
S2 Now wmax (17) wmin (5), Go To S3.
Comp Xi 1 2 3 4 5
Capacity, wi
2 3 3 4 5
5
1max 17
iiww
S3If (wi – wmin) i.e. 5 – 5 0
B = B X1 B = X1
A = A – X1 A = {X2, X3, X4, X5}
wmax = wmax – w1 wmax = 12 w = {4 3 3 2}Increment i i.e. i.j = 2, go to S2.
S2(wmax – wmin) = 12 – 4 = 8 go to S3.
S3(w2 – wmin) = 4 – 5 < 0 go to S4.
S4A (I) = X (J) X = {X2 X3 X4 X5}Do for I = i to n – 1, J = i + 1 to n begin
R = wmin – w2 e.2 = 5 -3 & R1 = Xj +1 – R R1 = 3 - 2 = 1
B = X1 X2 X3 end. Repeated the process for j = 3. For j = n – 1 (= 4)
B = B X2 X4
{A} = {A} – X2 {A} = {X3 X4 X5}
wmax = wmax (12) – w2 (3) wmax = 9
w = w – w2 {w} = { 3 3 2}Increment i go to S2.
S4 A [I] = X(J) i.e. {A} = {X3 X4 X5}
{X} = {X3 X4 X5}
B = B X3 X4
B = B X3 X5
[A] = [A] – [X3] {A} = {X4 X5}
wmax = wmax – w3 wmax = 8 – 3 = 5
w = {w} – {w2} {w} = { X4, X5}Increment i go to S2.
S2 this S4 are valid.
{B} = {B} X4 X5
S5 Set {A} is empty.
B ={X1X2X3X2X4X2X5X3X4X3X5X4X5}
S6 Bdisj=
S7 FR (k, n)= p1+q1p2p3+q1q3p2p4+q1q3q4p2p5+ q1q2q4p3p5+ q1q2q5p3p4+ q1q2q3p4p5
543215432154321
5432143213211
XXXXXXXXXXXXXXX
XXXXXXXXXXXXX
Reliability Evaluation of Complex Systems
Complex system Reliability Evaluation
• Probability calculus
• Event space
• Path set
• Cut set
• Decomposition
• Flow Graph Method/ Inspection Method
RELIABILITY EVALUATION METHODS OF COMPLEX
SYSTEMS
Classification 1:•Non Path sets or Cut Sets (NPOC) approaches
•Path Sets or Cut Set (POC) approaches
Classification 2:Based on maintenance activity of the system 1)Maintained systems 2)Non-maintained systems
Path sets and Cut Sets approach
1. In this approach, reliability evaluation requires path sets /cut sets knowledge
2. Obtain path sets/cut sets of the system from information on system topology or logical interconnection of elements within the system
3. Obtain an algebraic expression of system reliability in symbolic form from path sets and cut sets
• Employ elemental reliability to obtain system reliability
• If there are m disjoint path sets then Rs = S1 – S2 + S3 - …………+ (-1)m-1 Sm
where S1 ,S2 ,----- are sum of the probabilities of m path sets taken one at a time, two at a time,……m at a time
• It can also be represented as
s1 =
where Ti is the ith path set
Similarily
S2
Using sum of disjoint product (SDP) Rs can be expressed as:
P[ E1 U E2 U…….. Em ] = P[E1 ] + P[Ē1∩ E2 ] +P[Ē1
∩ Ē1 ∩ E3 ] + ……. + P[Ē1 ∩ Ē2 ∩ Ē3 ∩…….. ĒM-
1 ∩ Em]
NON PATH SETS OR CUT SETS APPROACHES
• The following methods belongs to the non path sets or cut sets approach of reliability evaluation
»Event space method»Decomposition method (Baye’s
theorem)»Method of inspection»Flow graph method»Probability calculus method
Event space method
• Assumed that each element and system has two states
» Operating and failed states
• This method is one of the most elementary and a straight forward method
• If a network has n elements, then there are 2n possible combinations
• List all possible combinations , and sort out success states, and hence write system success function as union of these success states
• From the above figure we can write
the structure function of the system is given as
S = { E1 U E2 UE3 U E4 U E5U E6U E7U E8 UE10 UE11 UE12 U E14UE15UE16UE17U E26 }Now transforming ith element state without super bar as pi and ith element with super bar as qi , then the reliability becomes Rs = p1 p2 p3 p4 p5 + q1 p2 p3 p4 p5 + p1 q2 p3 p4 p5 +
p1p2q3p4p5+p1p2p3q4p5+p1p2p3p4q5+q1q2p3p4p5+q1p2q3p4p5+…………+q1q2q3p4p5+p1p2q3q4q5
DECOMPOSITION METHOD
Baye’s theorem : if A is an event which depends upon of two mutually exclusive events Bi and Bj , of which one must necessarily occur, then the probability of occurrence of A is given by
P{A} = P{ A/Bi } + P{A/BJ } P{BJ }
RELIABILITY EVALUATION OF A BRIDGE NETWORK
METHOD OF INSPECTION• This method is useful for small systems with fewer branches,
and it is based on system configuration only
• In this method one has to trace out all sub graphs between the terminals for which the terminal reliability has to be computed
• Forward paths (without any loops) are computed first.
• Then paths with one loop, and subsequently two, three etc.
• Finally the system reliability would be the sum of product terms providing the product of reliability of involved branches and with an appropriate sign according to the sign rule
• Rs = F1 –F2 + F3 - ……………» Where F1 = Ui (all simple paths)
» F2 = Ui ( all sub graphs with one loop
» F3 = Ui (all sub graphs with two loops and so on)
• Mathematically reliability is given as
» Rs =
Sign rule is
Number of loops sign
o +
1 -
2 +
3 -
MARKOV PROCESS
• A Markov process is a stochastic process whose dynamic behavior is such that probability distributions of its future development depend only on the present state and not on how the process arrived in that state.
• In order to formulate Markov model ,first define all mutually exclusive states of the system
• The transition probabilities must obey the following two rules
» the probability of transition in time Δt from one state to another is given by z(t) Δt, where z(t) is the hazard associated with the two states . If all the z i(t) are constants the zi(t) = λI and the model is called homogeneous. If any hazards are time functions, the model is called non homogeneous.» The probabilities of more than one transition in time Δt are infinitesimals of a higher order and can be neglected.
Examples : one component system
• The probability of being in state S0 at time
t+Δt is written as P0 (t+ Δt ). This is given by the probability that the system is in state S0 at time t, P0 (t), times the probability of no failure in time Δt 1-z(t) Δt , plus the probability of being in state S1 at time t, P1 (t), times the probability of repair in time Δt, which equals zero
S0 = x1 and S1 = x’1
The resulting equation is
P0 (t+ t) =[(1-Z(t) t]P0 (t) + 0(P1 (t))△ △Similarly , the probability of being in state S1 at time t+ t is given by△
P1 (t+ t) = [z(t) t] P0 (t) + 1 ( P1 (t)△ △
The transition probability z(t) t is the probability of failure and the probability of△Remaining in state S1 is unity.
Re arranging the above equations we have
[P0 (t+ t)-P0 (t)]/ t = -z(t)P0 (t)△
[P1 (t+ t)-P1(t)]/ t = z(t)P0 (t)△ △
Passing to a limit as t becomes small, it was obtained as
dP0 (t)/dt = -z(t)P0 (t)
dP1(t)/dt = z(t)P0 (t)
The above equations can be solved in conjunction with the appropriate initial Conditions for P0 (t) and P1(t) . The most common initial condition is that the system is good at t=0 i.e is P0 (t=0) =1 and P1(t=0) =0
the solution is given as P0 (t) = exp[- ]
P1 (t) = 1- exp[- ]
MARKOV GRAPH
1-Z(t) △t 1
z(t) t△S0 S1
Markov model for two element system
• Let the two elements of the system be x1 and x2 and the corresponding four states will be given as
» S0 = x1 x2
» S1 = x’1 x2
» S3 = x1 x’2» S4 = x’1 x’2
• The probability expression for the state S0 is given by
» P0 (t+ t) =[(1-{Z △ 01(t) +Z 02(t)} t ]P△ 0 (t)
Where {Z 01(t) +Z 02(t)} t is the probability of a △transition in time t from S△ 0 to S1 or S2.
For state S1 ,
P1(t+ t) =Z △ 01(t) t P△ 0 (t) +[1-Z 13(t) t ]P△ 1(t)
Where Z 13(t) t is the probability of a transition from S△ 1 to S3 . Similarly for state S2 ,
P2(t+ t) =Z △ 02(t) t P△ 0 (t) +[1-Z 23(t) t ]P△ 2(t)
Where Z 23(t) t is the probability of transition from states △S2 to S3.
MARKOV GRAPH FOR TWO ELEMENT SYSTEM IS
S0 = X1 X2
Z 01(t)
Z 02(t) Δt
Z 13(t) Δt
Z 23(t)
Δt
S1 = X’1 X2
S2 = X1 X’2
S3 = X’1 X’2
ΔT
1-( Z01 (t) +Z02 (t)) ΔT
1-Z1 (t) ΔT
1-Z23 (t) ΔT
1
For state S3 the transition equation is
P3(t+ t) =Z 13(t) t P1 (t) + Z 23(t) t P2 (t) + 1 P3(t) △ △ △
RELIBILITY OF FLOW NETWORKS
RELIBILITY OF FLOW NETWORKS
ST Reliability: Most common quantitative index.
It is the probability of successful communication between
source and terminal node.
Basic Assumption: The required amount of flow can always
be transmitted from source to terminal node whenever a path
is available.
This Assumption: An implication which is neither physically
valid nor economically justified. Since most of the practical
system are flow limited networks.
Examples:
• Power Transmission system has limited power ratings of its power lines.
• Transport system might not allow traffic more than a particular value.
• A chemical system in which oil or gas flow through pipes is permissible only upto safe limits.
• A communication system having fixed channel capacities of its links.
Reliability of a flow Network:
The network is successful if and only if it is able to transfer at least the required flow from the input node to the out node.
The performance measure combining both reliability and capacity has been referred to as capacity related reliability (CRR) capacity constrained reliability (CCR).
Two types of performance indices (PIs) exist in literature.
(1)Weighted Reliability PI, PI-1 [30, 89]
It is defined as PI - 1
Where wi is the normalized weight & is defined as:
wi = Ci / Cmax
& Ri probability of the system being in state Si
& PI-2 [26, 28, 32, 35, 36]
The network is good if and only if a specified amount of flow (wmin) can be transmitted from the input node to output node or (s, t) node pair [36].
Si
iiRw
PI-1: More generalized.Aggarwal [30] State Enumeration ApproachKumar [89] Substitutionary Decomposition.Limitations:• Aggarwal’s Method [30] suffers from the fact that 2n system states
are to be evaluated. Thus rendering it unattractive for medium & large networks.
• It generates both cancelling and non-cancelling terms.• The PI contributing terms are required to be subjected to checks
to confirm if these contribute to PI or not.• Kumar’s Method [89] also generates both success and failed
terms but has the advantage over Aggarwal’s method that success terms are not subjected to such checks as in [30].
• The method generates large number of intermediate terms which are not included in the PI. Thus rendering the method computationally inefficient.
Proposed Method is free from these limitations.
PRELIMINARIES
Composite Performance Index (CPI) [30]
The weighted reliability measure i.e. composite performance index (CPI), integrating both capacity and reliability may be stated as:
Where wi is the normalized weight & is defined as:
wi = Ci / Cmax
i.e. the ratio of capacity in the i-th state to the maximum capacity (Cmax) of the system
& Ri probability of the system being in state Si & is given as:
)(xSi
iiRwCPI
1/ 0/ij ikSj Sk
kjiri qpXPR
Capacity Functions of Simple Networks [35]
For Series Network,
For Parallel Network
For non-series-parallel network:
Using max-flow min-cut theorem.
Key-Element:
A key-element is the branch of the network used to factor an expression into two expressions. One expression is obtained by substituting the key-element ‘1’ and the second expression is obtained by substituting the key-element “0” in the expression to be decomposed.
iSER CXC min
Xi
iPAR CXC
iallforcutww iiminmax
Let a success expression be given as below:
Selecting Xi as the key-element, the two
decomposed expressions obtained by substituting the
key element Xi ‘1’ and substituting ‘0’ are obtained as:
5325414321 XXXXXXXXXXS
532544321 XXXXXXXXXS
532431 XXXXXXS
53544321 1 XXXXXXXXS
544321 XXXXXXS
Final Term:
The key- term associated with S (key) that represents a decomposed expression in which each product term (belonging to sum of the products success expression) has been substituted by either one or zero. The right hand side of a final term contains no logical / Boolean variable either in complemented or in uncomplemented form.
…(1)
S (key) = 1 is a success term.
Terms for which S (key) = 0, say (1), has been termed as drop terms in [89]. These drop terms are not generated in the proposed method.
111154321 XXXXXS
054321 XXXXXS
54321 XXXXXS
Minimization of a decomposed expression (right hand side of a key-term-say represented by a S(key) in [89] has been done using the following Boolean algebra rules:
In the proposed method additional simplified algebra rules have been suggested. With the use of these rules, on the terms of decomposed expression results in faster generation of final term(s).Algebra RulesRule 1: … (2)
Provided no variable between Xi+1 to Xm is missing then (2) can be expressed as
Example: Let Then according to Rule 1
The Rule 1 is also applicable in case of variables in the key-term of S(key) are in complimented form and/or in the mixed form.
111;101;110;000 XXX &
m
ikki XXXS
11,...
1...... 2121 miii XXXXXXS
54321 XXXXXS
154321 XXXXXS
Rule 2: … (3)then the different resulting key-terms can be obtained as:
simplifying the s(key)’s are expressed as:
Example: Let then according to Rule 2 the resulting key-terms obtained
are given as follows:
The procedure of minimization using Kumar’s method [89] is given as below:
i.e. the procedure adopted using method [89] not only inefficient but generates drop terms also where as using any of the proposed rules no drop terms are generated.
211,..., iii XXXXS
2121211... iiiiiii XXXXXXXX
1... 211 iii XXXXS 1... 211 iii XXXXS
1... 211 iii XXXXS
54321 XXXXXS
154321 XXXXXS 154321 XXXXXS 154321 XXXXXS
54321 XXXXXS 54321 1 XXXXXS
54321 XXXXXS 11154321 XXXXXS 154321 XXXXXS
154321 XXXXXS
054321 XXXXXS
Rule 3: then the different resulting key-terms
generated are given as follows:
Illustration Rule 3:Letthen the key-terms generated are:
&
2
1
1...,i
ixki XXS
1... 1 iiXXS
1... 21 iii XXXS
1... 21 iii XXXS
54321 1 XXXXXS
14321 XXXXS
154321 XXXXXS
154321 XXXXXS
nnn XXXS 23...,
nnnnnnn XXXXXXXSskeyS 12123...'
nnnnnnnn XXXXXXXXS 123123....
1... 123 nnnn XXXXS
1... 123 nnnn XXXXS
Rule 4: Given
then success key- terms are generated using:
i.e. two S(key)’s are obtained & these are:
Illustration Rule 4:
Let
Two key – terms generated using Rule 4 are:
754321 XXXXXXS
17654321 XXXXXXXS
17654321 XXXXXXXS
Rule 5: Given The different (three) key – terms generated are:
Illustration Rule 5:Let The three key – terms generated using Rule 5 are given as below:
It has been observed that Boolean algebra rules X 1=1 and X XY = X yield incorrect results and should not be used in minimization process of the decomposing terms [89].
Rule 6: The key term obtained for a chosen key-element turns out to be a s-t cut then the term is not considered for further expansion.
3221... iiiii XXXXXS
1... 321 iiii XXXXS
1... 321 iiii XXXXS
1... 321 iiii XXXXS
76654321 XXXXXXXXS
17654321 XXXXXXXS
17654321 XXXXXXXS
17654321 XXXXXXXS
ALGORITHMThe algorithm is a modification over algorithm in [89]
S1) Let Pi, I = 1 to n be the minimal paths. These can be
obtained using any of the existing technique.
S2) Write down system success function (SSF),S expressed as union of minimal paths i.e.
S3) The network branches represented as logical variables are assigned numbers in a sequence (say X1 1, X2 2 & so
on). Let the number of variables be m.
S4) j 1
S5) Choose link / element j as a key-element
S6) Decompose expression in S2 into two “1” substituted and “0” substituted expressions for the chosen key-element j.
S7) Apply Rule 1 through Rules 6 (whichever applicable).
If or . It is a success term.
n
iiTSSSF
1
,
1jS 1jS
S8) if j > m then goto S 11.
S9) For the remaining S(key) terms (which do not result into retained terms): (a) check for the applicability of
minimization rules & / or (b) next variable (branch) be chosen as key-element.
S10) Repeat the process, S6 through S9 until all the success (retain) terms have been obtained.
S11) For each success term, obtain Ci using min-cut-max
flow theorem [36].
S12) The CPI for the system is obtain by replacing the logical variables to the corresponding probability variables i.e.
Xi pi
Xj pj
Si Skk
Sjii CqpCCPI
ijij
max//
/
11
ILLUSTRATIONS Example 3.4.1 CPI of bridge network shown in Fig. is to be evaluated. The number in parenthesis is the capacity of the element.
Figure
3
x2(3)
t s
x4(4)
x1(7)
x3(3)
x5(5)
4
(a)
Cmax = 7
E(5) E(5
Pr {E} = Pr {E} = PE
(b)
4
3
Solution: Generation of System Success Function (SSF), S
S1) The four minimal paths / Tie sets for the network are:
T1 = X1 X2 T3 = X1 X4 X5
T2 = X3 X4 T4 = X2 X3 X5
S2)
…(1)
S3) Number the links (elements)
X1 1, X2 2, X3 3, X4 4, X5 5
S4) j 1
S6) a) Substituting X1 1 in SSF(S2), we get
…(2)
b) is obtained by substituting X1 0 i.e.
…(3)
4321
4
1
, TTTTTSSSFi
i
5325414321 XXXXXXXXXX
532544321 XXXXXXXXXS
532431 XXXXXXS
1XS
S7) Since neither nor , none of these result in success term.
S8) j m (m = 5), algorithm goes to S9.
S9) Select X2 as a key element. Obtain decomposed expressions substituting X2 1, X2 0 .
The resulting four expression are …(4-7)
Apply Rule 1 to (7) the key-terms generated are expressed as follow: Since
…(8-9)then
53544321 1 XXXXXXXXS
544321 XXXXXXS
534321 XXXXXXS
4321 XXXXS
4321 XXXXS
14321 XXXXS
Apply Rule 5 to (5), three key terms are:
…(10-12)
S10) As all the success terms are not obtained, the process is repeated. i.e. Algorithm branches to S8.
As j ( = 2) > m ( = 5). Steps S9 through S11 are executed repeatedly until j > m. Performing S9 through S11, the following key – terms are obtained. From (4),
…(13-14)and (6)
…(15)is not expanded as is a cut (Rule 6).
154321 XXXXXS
154321 XXXXXS
154321 XXXXXS
5544321 1 XXXXXXXS
54321 1 XXXXXS
54321 XXXXXS 321 XXXS
31XX
From (13)
…(16-17)
From (14) Using Rule 3
…(18-20)
From (15) Using Rule 2
Applying Rule 3 on (16) & (17) the key-terms obtained are:
S11) The S(Key) Success terms generated alongwith capacity of each term are listed in Table.
5554321 111 XXXXXXXS
54321 1 XXXXXS
14321 XXXXS 154321 XXXXXS
154321 XXXXXS
154321 XXXXXS 154321 XXXXXS
154321 XXXXXS
154321 XXXXXS 154321 XXXXXS
154321 XXXXXS 154321 XXXXXS
TABLE
Sr. No.
Key terms Capacity Sr. No.
Key terms Capacity
1. 4 8. 4
2. 4 9. 3
3. 4 10. 4
4. 5 11. 7
5. 3 12. 7
6. 3 13. 3
7. 7 14. 3
4321 XXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
54321 XXXXXS
S12) The CPI may be expressed as given below:
The CPI defined is a general in nature as it may be used to evaluate the s-t reliability (connectivity measure), when all the links and the network is assumed to transport / communicate any amount of flow.
It can also be used to evaluate the PI as defined in methods [26, 89]. If Cmax appearing in the denominator of PI is dropped and the key
(success) – terms with capacity, C Cr, appearing in the expression of
PI are retained & rest of the terms (C < Cr) are dropped.
The proposed method generates 14 success terms, 11 intermediate terms and generation of failed terms is fully avoided whereas method [89] generates total 38 terms against 25 terms generated by the proposed method, 17 intermediate terms and 14 success terms. There is a total saving 34% with the proposed method.
7/}3
3774
7334
444{
max
Cqqrrr
rqrrrqrrrrrrrrrrrrrq
rrqrrqrrqrrrqrrrrqqr
qrrqrrrrqrrrqqCPI
edcba
edcbaedcbaedcbaedcba
edcbacddcbadcbaedcba
edcbaedcbadcba
• In life cycle cost analysis, the quality and reliability costs can be split into two components
» Controllable costs» Resultant costs
Controllable costs :
these are the costs of the activities that are planned and include all such activities which are necessary to ensure quality and reliability.
e.g. : Inspection costs, testing costs
• Resultant costs :» These are unplanned costs, and as such result from not
achieving the desired levels of quality and reliability.» These include external and internal failure costsThe manufacturer ha to include the following elements , to
optimize the costs and customer satisfaction,
Reliability design costs:1) planning costs2) Inspection and life
testing3) training and
management cost4) research and
development
Reliability section
• Responsibilities are
» Reliability data management and analysis» Reliability analysis and prediction» Reliability appointment and design» Specifications, material section and vendor control» Reliability tests planning and demonstration» Reliability education and statistical services» Intra departmental coordination.
Reliability an inter disciplinary effortPHASE EXPERT NEEDED
CONCEPTUAL SYSTEM ENGINEER
DESIGN AND DEVELOPEMENT MATHEMATICIANS OR SYSTEM ANALYSTS,DESIGN ENGINEER,PHYSICISTS,MATERIAL SCIENTIST,METTALURGIST,PROCESS TECHNOLOGIST,CHEMICAL ENGINEER,ELECTRICAL,ELECTRONICS,MECHANICAL,STRUCTURAL ENGINEER
MANUFACTURE AND INSTALL PRUCHASE ENGINEER,PRODUCTION ENGINEER,STATISTICIAN,QUALITY CONTROL ENGINEER,PACAKAGING EXPERT,MANAGEMENT EXPERTS,SERVICE ENGINEER
USE OPERATION ENGINEER,MAINTENANCE ENGINEER
Experts required to handle reliability problems
CHALLENGES FOR FUTURE
RELIABILITY MANAGEMENT
• It is concerned with the top management
• The concern of reliability management is to achieve organized reliability
• Any organization, to remain in business and to maximize customer satisfaction at the same time, has to look into the cost effectiveness of the activity i.e life cycle costs are important.
Methodologies for dealing with uncertainty , ambiguity, fuzziness and imprecision
CLASS OF AMBIGUITY METHODOLOGY USED
Uncertainty / Randomness
Theory of stochastic process and theory of decision making, Principle of uncertainty
Fuzziness Fuzzy set theory/ subjective probability theory/ theory of fuzzy or approximate reasoning/ invocation of knowledge engineering oriented approach
Ambiguity Fuzzy set theory/ fuzzy logic/ modal logic/semantics of information
Blur /Vagueness Filtering techniques /relaxation operations/ image interpretation techniques
Imprecision Structural modeling technique
Failure characteristics
Phases in equipment life
CONCEPT OF DEFINITION
DESIGN AND DEVELOPMENT
MANUFACTURE ANDINSTALL
OPERATION ANDMAINTAINANCE
BIRTH
QUALITY
MAINTAINABILITY
DEATH
RELIABILITY
HIGH PRESSURE OXYGEN SUPPLY SYSTEM OF A SPACECRAFT LIFE SUPPORT SYSTEM
Fault Tree Analysis
By
By Dr. G L PahujaNational Institute of Technology,
Kurukshetra
June 09,2009 GNEC Ludhiana
Fault Tree Analysis
• Fault Tree Analysis (FTA) is a systems engineering technique which provides an organized, illustrative approach to the identification of high risk areas.
• A Fault Tree (FT) is an event logic diagram, providing a logical representation of the events occurring within a complex system.
• Construction of a FT begins with the definition of the top undesired event (the system failure).
• The causes are then indicated and connected to the top event by conventional logic gates.
• The procedure is repeated for each of the causes and the causes of the causes, etc., until all the events have been considered.
Fault Tree
Product Development Chart with Scheduled FTA Inputs
Conceptual Phase
System Development
Phase
Equipment Development
Phase
Production Phase
Operational Use Phase
Feasibility Studies operational and
logistics concepts. System Analysis, Optimization, Synthesis
and Definition.
Detailed Equipment Design Layouts, Parts Lists,
Drawings, Support Data
Fabrication, Assembly, Test Inspect, Deploy,
Operational Equipment.
Operate and Maintain Equipment
in the Field
Conceptual Design Review
System Design Reviews
Equipment Design Reviews
Critical Design Review
In-service Design Review
Major Steps of Fault Tree Analysis
1. Define System and its bounds.
2. Define the undesired event (TOPEVENT).
3. Construct Fault Tree.
4. Perform qualitative evaluation.
5. Perform quantitative evaluation.
General Structure of a Fault Tree
Top Event:
System Failure
Resultant Events
AND/OR Gates
Basic Events
FT Construction Steps
1. Understand the system to be evaluated.2. Define the undesired event.3. Analyze the system to determine the logical
interrelationships of higher and lower functional events which may cause a predefined system fault condition.
4. Apply logical relationships to input fault events which are defined in terms of basic,independent and identifiable faults that may be assigned probability values.
5. Use FT symbols to connect this information.6. Reduce the FT if possible.7. Eliminate any feedback paths.8. Check to ensure all FT rules have been followed.
Symbol Symbol Name
Description Reliability Model Inputs
Basic Event
Basic event for which reliability information is available
Component failure mode, or a failure mode cause
0
Conditional Event
Event that is a condition of occurrence of another event when both must occurs for the output to occur
Occurrence of event that must occur for another event to occur
0
Event Symbols
Undeveloped Event
A part of the system that yet has to be developed-defined
A contributor to the probability of failure. Structure of that system part is not yet defined
0
Dormant Event
A basic event that represents a dormant failure
Dormant component failure mode or dormant failure cause
0
House External Event or Trigger Event
Occurrence of event that must occur for resultant Event to occur
0
Intermediate event
Event resulting from occurrence of other events
Consequent to other events
1
Gate SymbolsSymbol Symbol
NameDescription Reliability Model Inputs
AND Gate
The output event takes place if all of the input events occur
Parallel redundancy, one out of n equal or different branches
>2
OR GATE This output event occurs if any of its input event occur
Failure occurs if any of the parts of that system fails-series system
>2
PRIORITY AND
The output event occurs only if the input events do in sequence from left to right
Good for representation of secondary failures & enabling sequence of events
>2
INHIBIT GATE
The output occurs only if both of the input events do, one of them conditional
Conditional probability of occurrence of the final event
2
NOT GATE
The outcome is present only if the input event does not occur
Exclusive events or preventive measure does not take place
1
m
MAJORITY VOTE GATE
This output occurs if m of the inputs occur
Redundancy k out of n, where m=n-k+1
>3
EXCLUSIVE OR
The output event takes place if one, but not the other input occur
A failure of the system occurring only if one, not both of the two possible failures happens
2
Transfer Symbols
In
Out
Symbol Symbol Name
Description Reliability Model Inputs
Transfer Gate
Gate indicating that this part of the system is developed in another part or page of the diagram
A partial reliability block diagram that is shown in other location of the overall system block diagram
0/1
An example of the use of AND/OR gates..
Tank ruptures
Over Pressure
OR
Wall Fatiguefailure
Fig (a):
Over Pressure
AND
Excessive temperature
Relief Valvefails
Fig (b):
An example of the use of AND/OR gates..
Tank ruptures
Over pressure Wall Fatiguefailure
AND
Excessive temperature
Relief Valvefails
OR
Fig (c):
An R-C Filter Circuit
Input~ Output
X2
X1
X3
System States
T1 Normal output
T2 False output signal but safe
T3 False and dangerous output signal
States of R-C filter circuitComponent State
Capacitor: X1X11 Working Normally
X12 Open Circuit Fault
X13 Short Circuit Fault
Diode: X2X21 Working Normally
X22 Open Circuit Fault
X23 Short Circuit Fault
Resistor: X3X31 Working Normally
X32 Open Circuit Fault
Fault Tree for T1
T1
X11 X21X31
Fault Tree for T2
T2
X12 X13 X31 X13 X23X21
Fault Tree for T3
T3
X12 X31 X11 X23 X31X23
PARTS OF SOLDERING IRON 1. Cord Set (a) Plug (b) Cord 2. Handle 3. Heating Element 4. Soldering Tip 5. Shroud 6. Fiber Insulator 7. Solderless Crimp Connector 8. Long Screws 9. Short Screws (3) 10. Washers under Long Screws(2) 11. Nameplate
Critical Shock Hazard
Potential Shock caused by Defective Plug
Potential Shock caused by defective line cord
Potential Shock caused by shorted tip
Person grounded or at different
potential than shock
voltage
Person touches plug
The plug is connected
Person touches
defective cord or touch
something conductive at defect
Tip in contact
with high potential energy source
Power on condition
A
B C
D
Fault Tree for Soldering
Iron
Person in electrical contact
Insulation broken off
plug
Defective cord
Short in contact
with metal part
Plug cracks
Burned line cord
Abraded line cord Potential short
AC line cord to shroud or tip
Heater wire short
Hot tip or shroud contacts line cord
Line cord insulation not designed to withstand
foreseeable heatLine cord fiber
separator or insulator wire
failure
Person touches shroud
Person cleans
tip
Person adjusts screws
Cord rubs against rough
surface
Person touches conductive
surface
Tip in contact with conductive
surface
Line cord insulation not designed to
withstand foreseeable abrasion
Plug/Line cord
separation
A
BC
D
Plug is defective
Line cord defect
Flaw in tip
FTA-Simplification Rules
1. A = AO. A = OA . A = OA + A = 1A . A = AA + AB = AA + AB = A + B
A1 A2 = O
FTA – Example (MOCUS)T
I2I1
I4I3 X4
X1 X2
X31
X1 X32
FTA – Example (MOCUS)
T
FTA – Example (MOCUS)
T
I1
I2
FTA – Example (MOCUS)
T
I1
I2
X31I3
X4I4
FTA – Example (MOCUS)
T
I1
I2
X31I3
X4I4
X31X1
X31X2
X4X1
X4X32
FTA – Example (MOCUS)
T
I1
I2
X31I3
X4I4
X31X1
X31X2
X4X1
X4X32
T= X1X31 + X2X31 + X1X4 + X32X4
Uses of Fault Tree Analysis
1. Direct the analysis to ferret out failures deductively.
2. Pointing out the aspects of the system involved in the failure of interest.
3. Providing a graphical aid for those in system management who are not involved/removed from the system’s design changes.
4. Providing options for qualitative, as well as quantitative, system reliability analysis.
5. Allowing the analyst to concentrate on one particular system failure at a time.
6. Providing the analyst with insight into system behavior.
Fault Tree With Disjoint Events
Advantages of Fault Tree Analysis
1. Analysis to the degree of details desired.
2. Permits concentration on a particular undesired event at a time.
3. Important aspects with respect to failure event of interest identified.
4. Provides genuine insight into the system behavior.
5. Provides both qualitative and quantitative analysis.
6. Environmental and other external influences can be easily accommodated.
7. Easily modifiable to account for certain factors.
8. Provides visual and graphical aid for system management and planning.
Disadvantages of Fault Tree Analysis
1. Cost of development is high.
2. Shortage of skilled persons in the art of fault tree development.
3. Lack of efficient mathematical techniques.
4. Scarcity of pertinent failure data.
Current Literature in Reliability & Maintainability
IEEE transactions on Reliability Proceedings annual Reliability and
Maintainability symposium Technometrics Applied statistics Operations research IIE transactions Journal of the American statistical association
Reliability review Naval research logistics International journal of reliability ,quality and safety
engineering Microelectronics and reliability Reliability engineering Journal of applied reliability
ACKNOWLEDGEMENT
• Mr. Sreenivas Anchuri
• Ms. Deepika Arora
• Dr. J.S. Lather
• Dr. Krishan Gopal
• Mr.M.Vivekanand
• Mr.Manoj Gubbala
THANK YOU
