Transcript Stochastic Studies on Deteriorating Systems: Optimal

Studies on a New Class of Shock
Models for Deteriorating Systems
Prof. Alagar RANGAN
Ayşe Tansu TUNÇBİLEK
Department of Industrial Engineering, Eastern
Mediterranean University, North Cyprus
1
Outline
•
•
•
•
•
•
•
•
Introduction
 - Shock Models
The Stochastic Model
The Statistical Characteristics
Specific Models and Discussions
Conjecture
Simulation and Estimation
Optimal Replacement Model for A NonRepairable System
2
Outline
• Optimal Replacement Model for A Repairable
System
• Optimal Replacement Model for A NonRepairable System and Spare with Lead Time
• Optimal Replacement Model for A NonRepairable System with Emergency Replacement
• Optimal Replacement Model for A Non-Repairable
System and Spare with no Lead Time
• Applications
3
Introduction
• All real world systems are deteriorating in
nature. Majority of such models aproach
the problem through shock models.
Abdel Hameed (1986), Feldman (1976), Shanthikumar (1983)
• A system is subject to randomly occuring
shocks, each of which adds a nonnegative
random quantity to the accumulated
damage process.
4
Shock Models
• The term shock refers any perturbation to
the system.
The system fails when
accumulated damages crosses a
threshold value
Cumulative
damage

threshold value
series of randomly
occurring shocks
Y
X
repair time
Lethal shock
time
working time
5
• Yeh and Zhang (2004) in a refreshing departure
introduced a new class of shock models and
called them  - shock models.
• Earlier shock models concentrated solely on the
magnitude of the damage caused by the
shocks, Yeh and Zhang’s model paid attention
to the frequency of the shocks.
6
 -Shock Models
• A system is likely to fail if two successive shocks
occur within a short time, whereas it may not fail if
they are separated by a longer duration.
As an example
• Think of an elastic material which is subjected to
stretching.
• If the second stretching occurs before the material
recovers from the first stretching, the material breaks.
Lethal shock
0
T
7
The Stochastic Model
• This study attempts a comprehensive analysis of
 - shock models in which the shock counting
process is generalized to a renewal process, and
• Threshold times are considered as random
variables.
• Apart from deriving explicitly various statistical
characteristics of the model we analyze optimal
replacement problems of such a system.
8
Notations Used
• Z: Random variable denoting the time between two
successive shocks with pdf f Z ..
• D: Random variable denoting the threshold value
with pdf g D . .
• W: Random variable denoting time between two
successive failures with pdf kW t .
• N (t ) : Counting variable denoting the number of
failures in (0, t ) .
• M (t )  E N (t ).
• L f (s) :Laplace Transform of the density function f.
9
I.The Model Assumptions
Assumption 1
The system is subject to shocks. The time
between shocks, are assumed to be
independently and identically distributed.
10
Cycle 1
Cycle 2
Fails
>
Cycle N
Fails
<
Time between two shocks >

nonlethal shock
Time between two shocks <

lethal shock
11
Assumption 3
• Threshold value  is a random variable.
Assumption 4
• The shock arrival times and the threshold
value times are independent of each other.
12
The Statistical Characteristics
• Probability density of W can be represented as the
sum of a random number of random variables,
N 1
W   X i  YN
i 1
time between two
successive failures
•
X i : a sequence of independently and identically
distributed random variables, which are distributed as
Z but conditional on Z>D.
•
YN : random variable distributed as Z but conditional
on Z≤D. It is assumed to be independent of the
sequence X i’s.
13
• N has the geometric distribution
P  N  n  qpn , n  0,1,2,...
where
q  P[Z  D]
• Define the conditional distributions of X i and YN
as
f Z x GD x 
 x   Px  Z  x  dx | Z  D 
PZ  D
and
f Z ( x)GD ( x)
 ( x)  Px  Z  x  dx | Z  D 
P(Z  D)
14

kW (t )    *  (t ) P(Z  D)  P  Z  D 
(n)
n
n 0
• And taking Laplace Transform of kW (t );
Lk ( s ) 
L fG ( s )
1  L fG ( s )
• Lk (s) could alternatively be derived by
writing the integral equation
t
kW t   f Z t GD t    f Z  GD  kW t   dt
0
15
0
0
t
The first shock itself
occurs only after t
P(W  t )

t
In the remaining interval  , t 
of length t    there is no
failure
t
K W t   FZ t    f Z  GD  KW t   d
0
The first shock occurs at
some instant   0, t 
The threshold time starting
from t=0 is over by time 
16
•
With simple differentiation;
t
kW t   f Z t GD t    f Z  GD  kW t   dt
0
•
Application of Laplace Transform we get the
Lk ( s ) 
•
L fG ( s )
1  L fG ( s )
Abate, J and Whitt W (1995),”Numerical
Inversion of Laplace Transforms of Probability
Distributions”, Informs Journal on Computing,
7, 36-43.
17
• The Mean and Variance are obtained after
simplifications;
EW  
E Z 
PZ  D 
 
E Z2
2 E Z E Z | Z  D PZ  D   E 2 Z 
VarW  

2
PZ  D 
PZ  D 
18
Specific Models and Discussions
• When the system is subjected to the same kind
of shock each time, the threshold time of the
system is likely to remain a constant, a case
discussed by Yeh (2004,2006).
• Under such a scenario, we have considered a
few models for different shock arrival
distributions:
– Exponential
– Gamma
– Uniform
19
Conjecture
– For constant threshold times, amongst all the
nondegenerate shock arrival distributions with
common mean, exponential shock arrivals
lead to the minimum mean failure.
– Amongst all the shock arrival distributions with
common mean, the degenerate threshold time
distribution leads to the minimum mean
failure.
20
II. Simulation and Estimation
• Moment Estimators
• Deteriorating Systems have been widely modeled
using Power Law Process which is inhomogeneous
Poisson Process with intensity function
Scale parameter
t
 (t )   
  
 1
 , , t  0
Shape parameter
• We have fitted a Power Law Process to our model by
estimating ˆ and ˆ from our simulated failure times.
• Hypothesis Testing
ˆ
ˆ


• Bayesian Estimation to estimate
and
for various
apriori distributions.
21
III. Optimal Replacement Model for A Non-Repairable
System
C(T)
increase in operating
cost rate per unit with
successive shocks
2C
C
t1
t2
t3
tN (T ) T
• The total cost of running the system
aT  Ct2  t1   ... N T CT  t N T   c0
Cost of operating the
system/unit time
Cost of periodic
replacement
Period of replacement
of the system
22
• The long run expected cost of operating the
system per unit time;
T
E C1 T  
aT  C  E N t dt  c0
0
T
• The optimal value of the periodic replacement
time always exists and is given by the unique
solution of the integral equation
T
c0






M
T

M
t
dt


0
C
where M t  is the expected number of failure in 0, t  .
23
• All shock models in the literature have minimized
the expected cost per unit time to obtain the
optimal T*.
• By controlling the second order characteristics of
the failure process, the performance criterion of
the system is controlled, so cost minimization
emerges by considering variance of the number
of shocks in (0,T).
24
*
T
• 2 be the optimal period for replacement using
the cost function
EC2 T   ECT   1    VarCT 
• We have obtained optimal T and T for
*
C
(
T
vaious cases and compared the costs,
1 )
*
and C(T2 ) .
*
1
*
2
25
IV. Optimal Replacement Model for A Repairable
System
Cycle 1
Cycle 2
Fails
Cycle N
Fails
Fails
0
Replace
the item

>
X1

2

n
2
n
repair time
Y1
X2
Y2
time for the first failure
During the repair time, system is shut and arriving shocks have no effect on
the system
26
For our model we have derived C(N), and the Optimal N* is obtained
Expected successive repair times,form a
geometric process with rate b
Repair cost rate
c
C N  
N 1

n 1 b
N
n 1
 n  
n 1
N
1
 r  n  R
N 1

Replacement cost
n 1
1
n 1
b
n 1

Expected Random
Replacement time
Expected operating time of the system
following the (n-1)th repair in a cycle
•We have derived optimal N* and the conditions which N* uniquely exists.
27
V. Optimal Replacement for A
Non-Repairable System and Spare with Lead Time
• This study considers a generalized agereplacement policy of a system subject to shocks
with random lead time first considered of Sheu
and Chien, 2004.
Model:
1. In the age-replacement policy, planned (sheduled)
replacements occurs whenever an operating unit
reaches age T and a spare unit is available.
28
2. System failure is governed by our model
stated in I so that the distribution function
of faliure time is W(t).
3. A spare unit for replacement can be
delivered upon order and the Lead time L
has distribution function H(t).
29
The time between system replacements form a
cycle. The length of a cycle is calculated using
the following 4 mutually exclusive cases.
Case 1:
T
L<T<TF
0
L
T
TF
If the ordered spare arrives before time T and no failures occurs before T,
then delivered unit is put into stock and unit is replaced by that spare at age
T, at a cost  1
30
Case 2:
L
L<T<TF
0
T
L
TF
If the ordered spare arrives after time T and no failures occurs before the
arrival of the ordered spare unit, then the unit is replaced by a spare as
soon as the spare is delivered at a cost 2
31
Case 3:
TF
L<TF<T
0
L
TF
T
If the ordered spare arrives before failure which occurs before the time T,
then the delivered unit is put into stock and the unit replaced by the spare
upon the failure at a cost 3
32
Case 4:
L
TF<L
0
TF
L
If failure occurs before the arrival of the ordered spare, then the system is
shut down and replaced by the spare as soon as the spare is delivered at a
cost 4
-The cost rate for stocking is cs
-The cost rate resulting from system
down is cp
33
E[cost/cycle]
E[cost/unit time]=
E[ length of the cycle]
C (T ) 
T

0
0

T

0
0
(3   2 )  H (t )dW (t )   2  H (t )dW (t )   4  W (t )dH (t )  cs  H (t )W (t )dt  c p  W (t ) H (t )dt

0

 W (t ) H (t )dt   H (t )dt
Let
0
0

T
T
T

Q(T )    W (t ) H (t )dt   H (t )dt(3   2 )rW (T )  cs   (3  2 )  H (t ) w(t )dt  cs  H (t )W (t )dt
0
0
0
0

and



0
0
0
K  2  H (t ) w(t )dt  4  W (t )h(t )dt  c p  W (t ) H (t )dt
h(t ) : lead time density
w(t ) : failure time density
34
• We have established the following:
• Q(T ) is monotonically increasing and
Q(0)  K ,
Q()  K
T*  
Q(0)  K ,
Q()  K
T *  unique, finite
Q(0)  K ,
Q()  K
T*  0
35
VI.Optimal Replacement for A Non
Repairable System with Emergency Replacement
• In system V we considered the case when the system
could be shut down due to nonavailability of the spare.
However systems such as power distribution cannot be
shut down.
• When the spare is unavailable, we replace the system on
failure using emergency replacement. We assume the
cost of emergency replacement to be CE >> Cp. In this
case the expected cost rate is,
36

T
C (T ) 

T
0
0
(3   2 )  H (t )dW (t )   2  H (t )dW (t )  5  W (t )dH (t )  cs  H (t )W (t )dt
0

0

 W (t ) H (t )dt   H (t )dt
0
0
• A similar optimality analysis has been carried out
37
VII. Non-Repairable System and Spare
with no Lead Time
Failure
0
0

T

T
C (T ) 
Unscheduled cost c1
Scheduled cost c2
c1W (T )  c2 W (T )
T
W ( t ) dt
0
and
T


W (T ) c1  c2 {rw (T )  W (t )dt  W (T )}  c1 
0


C(T ) 
2
T

  W (T )dt 


0

38
Optimality Conditions
*
T
 .
c

c
1. If 1 2 , C (T ) is a decreasing function,then
2. If c1  c2 ,
T
Let
B(T )  c1  c2 {rw (T )  W (t )dt  W (T )}  c1
0
*
*
T
B
(
T
)0
The optimal
is given as the solution of
*
If B(T )  0
has no solution then T  
Further if rw (T ) is increasing which is a natural
assumption, it has been shown that B(T ) is an
*
T
increasing function, establishing the uniqueness of .
39
Optimality Conditions
• The condition for the existence of a finite
optimal T * ,
c1
rw () 
c1  c2 EW 
40
Application Areas
The developed model has several interesting
applications in diverse areas of which we will
mention a few:
i. Neurophysiology
ii. Stochastic Clearing Systems
iii. Fatigue Failure of Material
iv. Renewing Warranty Modeling
41
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Block H.W., Borges W.S. and Savits T.H., (1985) Age
dependent Minimal Repair, Journal of Applied
Probability, 22, 370-385.
Brown M., Proschan F., (1983) Imperfect Repair, Journal
of Applied Probability, 20, 851-859.
Kijima M., (1989) Some results for repairable systems
with General Repair, Journal of Applied Probability, 26,
89-102.
Abdel Hameed M., (1986) Optimal replacement of
systems subject to shocks, Journal of Applied
Probability, 23, 107-114.
Feldman R.M., (1976) Optimal replacement with semiMarkov shock models, Journal of Applied Probability,
13, 108-117.
Shanthikumar J.G. and Sumita U. (1983) General shock
model associated with correlated renewal sequences,
Journal of Applied Probability, 20, 600-614.
Nakagawa T., (1976) On a replacement problem of a
cumulative damage model, Operational Research, 27, No
4, 895-900.
Kijima M., Morimura H. and Suzuki Y., (1988) Periodical
replacement problem without assuming minimal repair,
European Journal of Operational Research, 37, 194-203.
Kijima M. and Nakagawa T., (1991) A cumulative damage
shock model with imperfect preventive maintenance,
Naval Research Logistics, 38, 145-156.
Stadje W., (1991) Optimal stopping in a cumulative
damage model, Operations Research Spectrum, 13, 3135.
Barlow R.E. and Hunter L., (1960) Optimum Preventive
Maintenance Policies, Operations Research, 8, 90-100.
Lam Yeh and Yuan Lin Zhang, (2003) A Geometric –
Process maintenance Model for a deteriorating system
under a Random Environment, IEEE Transactions on
Reliability, 52, 83-89.
Blischke W.R. and Murthy D.N.P., (1994) Warranty Cost
Analysis, Marcel Dekker, New York.
13.
14.
15.
16.
17.
18.
19.
20.
Blischke W.R. and Murthy D.N.P., (2000) Strategic
Warranty Management: A Life Cycle Approach, IEEE
Transactions on Engineering Management, 47, 4054.
Blischke W.R. and Scheuer E.M., (1998) Calculation
of the cost of warranty policies as a function of
estimated life distribution, Naval Research Logistics
Quarterly, 22 (4), 681-696.
Nguyen D.G. and Murthy D.N.P., (1984) Cost Analysis
of Warranty Policies, Naval Research Logistic
Quarterly, 31, 525-541.
Lam Yeh, Peggo Kwok W. and Lam (2001) An
extended warranty policy with option open to
consumers, European Journal of Operational
Research 131, 514-529.
Alagar Rangan, Dimple Thyagarajan and Sarada Y.
(2006) Optimal replacement of systems subject to
shocks and random threshold failure, Interantional
Journal of Quality and Reliability Management, Vol
23, 9, 1176-1191.
Lam Yeh and Yuan Lin Zhang (2004) A shock model
for the maintenance problem of a repairable system,
Computers and Operations Research, 31, 1807-1820.
Abate, J and Whitt W (1995). Numerical Inversion of
Laplace Transforms of Probability Distributions,
Informs Journal on Computing, 7, 36-43.
Abate, J and Whitt W (2006). A Unified Framework for
Numerically Inverting Laplace Transforms, Informs
Journal on Computing, 18, 408-421.
42
–
“Some Results on a New Class of Shock Models”
Asia-Pacific Journal of Operational Research
(APJOR) (SCI Expanded) Accepted
–
“A new shock model for systems subject to random
threshold failure”, A.Rangan, A.Tansu, International
Journal of Computer and Information Science and
Engineering 2;3,Summer 2008,p.203-208.
43
Thank You
44