An Inspection-Maintenance Model for a degraded system subject to 8-random shocks

CAl Jing, ZUO Hong-fu, ZHU Lei Civil Aviation College

Nanjing University of Aeronautics and Astronautics Nanjing, China

caijing@nuaa.edu.cn

Abstract-There are many degraded systems existed in the

complex system such as the aircraft, and it is very significant to plan appropriate Inspection-Maintenance activities to decrease the failure probability and maintenance cost rate. So an inspection policy for a degraded system with two independent competing failure processes is proposed firstly. Then a

degradation path and <5 - random shocks are adopted and

analyzed. Furthermore, the inspection interval (I), threshold coefficient (k) and PM critical threshold value for degradation (L) are chosen as optimized variables, and an unequal Inspection-Maintenance optimal model is developed to minimize the maintenance cost rate, and the solution method to model is given. Finally, a numerical example is illustrated for case study, and the result proves the model validity and effective.

Keywords- degraded system; inspection; maintenance model; <5 - random shocks

I. INTRODUCTION

It is well known that the effectiveness of a system depends on both the quality of its design and manufacturing process as well as the appropriate inspection-maintenance actions to prevent it from failing. In order to meet the needs of higher safety and reliability, condition-based maintenance refers to the practice of triggering maintenance activities as necessitated by the condition of the system. The condition of the system is acquired by parameters that are continuously monitored or periodically inspected. Because it is always costly or technically difficult to actualize continuously monitored, periodically inspection is always adopted.

For a degraded system, there are many adaptive existing inspection optimization models [1-5], but the random shocks and unequal inspection intervals are always not been considered in those models. Because there is a big difference between those assumptions and the situation of the actual degraded system inspection, an appropriate optimal inspection maintenance model for a degraded system should be developed to minimize the maintenance cost rate subject to a required reliability. In this model, the random shocks, degradation-rate variation and unequal inspection interval will be considered

Sponsors: Aviation Science Foundation (2009ZF52059) , National Natural Science Foundation of China (60939003), the special research project of NUAA ( NS2010171)

and researched; what's more, it is necessary to analyze the accuracy of reliability estimation.

II A POLICY FOR A DEGRADATION SYSTEM

A. System Description

The system is subjected to a variety of governing failure processes. We consider two independent competing failure processes in which one of them is degradation process, which is measured by Y(t), and the second is a random shock

process D(t) , whichever process occurs first causes the system to fail. And the system is not continuously monitored, its state can be detected only by inspection, but system failure is self-announcing without inspection.

B. Degradation Path

yet) = A + B . get) is called a random- coefficient

degradation path, where A > 0 and B > 0 are independent random variables and get) is an increasing time-dependent

function. The random variable A measures the initial value of degradation due to a different manufacturer, the manufacturing quality control of new items, variable deterioration during storage until the item is put into service, and so forth. Therefore, the initial degradation value A is a random variable. The variable B is the degradation rate and represents the variations among the population; get) is an increasing function.

c. t5 - Random Shocks

The so-called <5 - shock model is that the system will fail, if the time interval between two adjacent shocks is less than the parameters given in advance (or threshold). In other words, the model ignores the impact of the losses caused by a single shock, and just attaches importance to the impact of continuous shocks. Now, it is assumed that a repairable system is subject to the impact strength of the Poisson A process

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{ N(t), t > 0 }, and sl' S2' • • " S N(t) indicate arrival times

of the shocks, and xl'x2,",XN(t) indicate the intervals

between two continuous shocks; so we can get obviously,

Sn=X1+X2+" '+Xn , n=1,2,· · · ,N(t). When the

nth shock interval xn < 0 , the system will fail, i.e. the

th th (n -1) shock and the n shock together lead to system

failure. Because So = 0 , then XI = SI - So ' t = 0 is the

system start time, assuming that the system starting means a shock to the system. T is the working time before system

failure, so R2 (t) = Pz (T > t) is the system reliability

function. [t/8] [A(t - J' . 0)])

R (t) = e -At "" -=------� , 2 L... ., '

Where )=0 J.

t�O

[t /0] is the smallest integer not exceeding t /0 .

D. Policy Description

An unequal inspection policy is proposed in this paper as follows:

(l) The system is periodically inspected, and the first inspection (threshold) takes place at k· I time, then a repeated inspection takes place every I time units.

(2) When inspection, if the system degradation is not exceeding the setting state L, nothing is done; if the system degradation is during [L, G] preventive maintenance is done;

if the system degradation is exceeding the setting state G , corrective maintenance is done. Both preventive maintenance and corrective maintenance can restore the system operating condition to be as good as new.

(3) The random shock damage maybe lead to the system failure anytime, and if happened, the corrective maintenance is done.

(4) After a PM or a CM action is performed, the system is renewed. A new sequence of the inspection would start again, defmed in the same way.

I I I. OPTIMAL INSPECTION MODEL

A. Assumptions

To describe the characteristic of any real world systems to which the results of this paper may apply, the following assumptions are made:

(l) The system is not continuously monitored, its state can be detected only by inspection, but system failure is self­announcing without inspection.

(2) The two processes yet) and D(t) are independent.

(3) Since Yet) describes the total damage up to time t, it

is natural to assume that it is no decreasing. (4) A CM action is more costly than a PM, and a PM

costs much more than an inspection. This

impliesCc > C p > C[. We consider a degradable system suited at a random

environment where degradation and random shocks can contribute to an effect of the life of a system. we discuss the

case where systems are subject to two failure processes, called a continuous and increasing degradation process Y(t) , and

the a random shock process. Which ever process occurs first causes the system to failure.

B. Optimal Inspection Model

The objectives of the model are to determine the optimal PM threshold L and the optimal threshold inspection time k . I and repeated inspection time, in order to minimize the total maintenance cost rate. From the basics of renewal reward theory, an explicit expression for the average long-run maintenance cost per unit time can be derived [6-8]:

Where

EC(k,I,L) = lim C(t) = E[C]

Hro t E(T )

E[ C] is the expected total maintenance cost in a cycle;

E[T] is the expected life in a cycle.

C. Expected Total Maintenance Cost

The expected total maintenance cost during a cycle E[ C] can be expressed:

E[C] = C; · E[N] +Cp .pp +Cc'� Where

Ci : Inspection cost

C p : Preventive maintenance cost

Cc: Corrective maintenance cost

E[N] : The expected number of inspections during a

cycle.

Pp : The probability ofpM.

p.: : The probability of CM.

(l) Calculate E[ N]

E[ N] can be obtained in the following way: 00

E[N] = �>.�(N[ = i)

i=l

P( N[ = i) is the probability that there a total of i inspections occur in a renewal cycle.

00

Obviously, "LP(N[ = i) = 1. t=l

Accordingly to the policy, we can know that the

inspection will stop when the i-th inspection finds that a PM

condition is satisfied while this situation was not revealed in

the previous inspection, or the system fails during the interval

[ i . I 5, T 5, (i + 1) . I ] while the system is in the doing­

nothing zone before i · I . When i = 1 , we can get

P(N[ = 1) = �[L S; Y(k· J) S; G]· �(k· J) + �[Y(k·J) < L]·{I-�(k .J)}

When i � 2 , we can get

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And

P(Nf = i) = �{L � Y[(k+i-I)·I] � G} ·�{Y[(k+i -2 )· I] < L}· R2[(k + i-I)·I] +�{Y[(k+ i -2)·1] < L} ·�[(k+i -2)· I]

·P[(k+ i -2)·1 < T � (k +i -1)· I] Where

�{L � Y[(k+i -1)·1] � G}

={

¢[G -[,uA + ,uB(k + i-I)· I]] �(j� +(j�[(k+i-I).I]2

_ [L-[,uA +,uB(k+i-I)'I]]} ¢ I 2 2 2 -V (jA + (jA(k + i-I)· I]

� {Y[(k + i-I)· I] < L}

= {

¢[L -[,uA + ,uB(k + i-I)· I]]} ;

�(j� +(j�[(k+i-I).I]2

�[(k+i-I)·I]

. [[(k+i-]}f]/S) [1L([(k + i-I) . I] -J'O)]} ; = e -.<[(k+l-I)·f] L

}�O j!

P{(k+i-2 )·I <T�(k+i-2 )·I} = � {Y[(k + i -2 )· I] < L, Y[(k + i-I)· I] ;;::: G}

·�[(k+ i-I)· 1]+ �{Y[(k+ i-I)· I] � L} . {�[(k + i -2 )· I] -�[(k + i-I)· In

Since Y[(k+i-2 )·I], Y[(k+i-I)·I] are not

independent, we could obtain the joint pdf

!Y[(k+i-2}l],Y[(k+i-l}I](Yl'Y2) in order to compute

� {Y[(k + i -2)· J] < L, Y[(k + i-I)· J] ;::: G}.

We consider two different expressions for Y(t).

{Y1 = a +b((k+i -2)· J)

Y2 = a+b((k +i -1)· J)

After simultaneously solving the above equations in

terms of Yl andY2' we obtain:

a = Yl' g((k+i-I)·I)-Y2' g((k+i-2)·I)

g((k+ i -1)·1)-g((k+ i -2)· I)

= � (Yl'Y2)

b = Y2 -Yl

g((k+ i-I)· 1)-g((k+ i -2)·1)

= �(Yl'Y2) The Jacobian J is given by

8� 8�

J=�] 8yz 8hz 8hz 8y] �z

= Ig((k+i -2).1) � g((k+i -1)'1) 1 Then the random vector { Y[(k + i -2 )· I] ,

Y[ (k + i-I) . I] } has a joint continuous pdf as follows

IY[(k+i-2).!],Y[(k+i-I)-I] (Yl' Y2) =

IJI/A [h.(Yl'Y2) JIB [hz(Yl'Y2) J So

�{Y[(k+ i -2 )· I] < L,Y[(k+ i -1)·1];;::: G} ooL

= f flJIIA [�(Yl'Y2) ]IB [�(Yl'Y2) ] dy\dY2 GO

(2) Calculate � and �

Note that either a PM or CM action will end a renewal

cycle. In other words, these two events are mutually exclusive

at the renewal time point. As a consequence, Pp + � = 1 . The

probability P can be obtained as follows: p 00

Pp = L{�{Y[(k+i-2 ).I]<L, i�]

L � Y[(k +i -1)·/] < G} ·�[(k+ i-I)· I]} Where

�{Y[(k+i-2 )·I] < L,L � Y[(k+i-I)·I] < G} GL

= f ]JI/A [�(YPY2) ]IB [�(Yl'Y2) ] dy1dY2 L 0

D. Expected life

Since the renewal cycle ends either by a PM action with probability � or a CM action with probability �, the mean cycle length E[T] is calculated as follows:

Where

00

E[T] = L[(k+i-I).I] i=1

.{�{L � Y[(k+ i -1)·/] � G}

·�{Y[(k+ i -2)·1] < L}· �[(k +i -1)· I]} +� ·MI'TF

MTTF is the mean time to failure, As the defmition, we know

00

MI'TF = fR(t)d t o

So we can get

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00

MTTF = fp{T > t}dt o

m = 0 .Based on the figure 1 algorithm flow, we can get

the optimal values following as the table 1.

00

= f�[Y(t) s G]· R2(t)dt o

E. Ssolution Method to Model

To solve the optimal inspection interval I, the coefficient

k of threshold inspection and the PM threshold L, A step­

by-step algorithm proposed by Li and Pham [9) based on the

NeIder-Mead downhill simplex method is summarized as the

figure 1, where f is an average value.

IV. NUMERICAL EXAMPLES

Here we present an example to illustrate the results and

the step-by-step application procedure. Assume that the

degradation process is described by Yet) = A + Bg(t), both

follow normal distributions, with mean 50 and variance 5, and

mean 60 and variance 10, respectively. In short,

A � N(50,5) and A � N(60,10) . The degradation

function is assumed to be get) = t15 • Also G = 4000. And

assume that the reliability function for the random shock

damage is described by

Where

-At [tiS] [ A(t - j . 0) Y

R2(t)=e I ., j=O J.

A = 0.01, 0 = 30.

t�O,

And Ci = 100 , Cc = 800 , Cp = 400 .We now

determine both the values of k , I and L so that the

average total cost per unit time EC (k,I, L) is

minimized.

Since there are three decision variables k, I and L, we need (n+ 1) = 4 initial distinct vertices, which are

Z(I) = (2,15,2000) Z(2) = (1,30,1000) Z(3) = (3,10,3000) ,and Z(4) = (3,30,3000) Set

TABLE 1. OPTIMAL VALVES K, I AND L

m z(l) Z(2) Z(3) EC(2,15, EC(1,30, EC(3,1O,

o 2000)=60.9 1000)=73.5 3000)=90.7 EC(2,15, EC(1,30, EC(3,10, 2000)=60.9 1000)=73.5 3000)=90.7 EC(2,15, EC(2.5,24.2 , EC(1,30,

2 2000)=60.9 2500)=70.1 1000)=73.5

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Z(4) compute result

EC(3,30, EC(1,6.6, 3000)=120.9 1000)=100.6 EC(1,6.6, EC(2.5,24.2 , 1000)=100.6 2500)=70.1 EC(3,10, EC(2.42,18.2 , 3000)=90.7 2417)=83.5

2011 Prognostics & System Health Management Conference (PHM2011 Shenzhen)

EC(2,15, EC(2.5,24.2 , EC(1,30, EC(2.42,18.2 , EC(3,1O, 3

2000)=60.9 2500)=70.1 1000)=73.5 2417)=83.5 3000)=90.7 EC(2.25,19.6, EC(2.2 1,16.6, EC(2.48,14.4, EC(2,15, EC(1.5,22.5,

4 2250)=52.7 2209)=56.5 2479)=59.3 2000)=60.9 1500)=63.8

9 EC(2.42,2 1.8, 2653)=49.7

EC(2.40,2 1.7, 2650)=49.9

EC(2.38,2 1.6, 2654)=50.1

EC(2.44,2 1.5, 2 657)=50.1

Stop

Table I illustrates the process of the NeIder-Mead

algorithm. From Table 1, we observe that a set of the optimal

values is

k* = 2.42,1* = 2 1.8,t = 2 653 And the corresponding cost value is

EC* (2.42,2 1.8,2653)=49.7. From table 1, we can know that when the optimized

variable inspection interval (I ) and threshold coefficient (k ) are chosen, the higher PM critical threshold value for degradation ( L ) means that it is more likely to result in a failure, so there is a higher maintenance cost. When optimized variable I and L are chosen, the very high k will also put the system at high risk of failure. And also, frequent inspections will reduce the probability of failure, while incurring additional cost.

V. CONCLUSION

According to t two independent competing failure processes, an unequal Inspection-Maintenance optimal model is provided. In the model, the inspection interval ( I ), threshold coefficient (k) and PM critical threshold value for

degradation (L ) are chosen as optimized variables. However, the imperfect inspection used in the practice should be incorporated into the decision process formulation in the

future research. Finally, through the example, we can make a result:

(l )The threshold interval should be more than the repeated inspection interval for the degraded system with increasing failure rate.

(2)The well-timed PM is a pivotal to decrease the maintenance cost.

REFERENCES

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[3] Grall A, Dieulle L, Be'renguer C, Roussignol M. Continuous-time predictive-maintenance scheduling for a deteriorating system [J]. IEEE Transact Reliable 2002; 51(2):141-50.

[4] G. A. Klutke, Y. 1. Yang: The availability of inspected systems subjected to shocks and graceful degradation, IEEE Trans. Reliable. 2002,44,371-374.

[5] David F. Percy. Preventive Maintenance Models for Complex Systems. Springer Series in Reliability Engineering, 2008:179-207.

[6] Wang Guanjun, Zhang Yuanlin. 0 -Shock Model and the Optimal Replacement Policy. JOURNAL OF SOUTHEAST UNIVERSITY (Natural Science Edition). 2001, 5(31): 121-124.

[7] Chao Chien-Min. A Study on Modeling Functional Inspection at Two Stages [J]. SYSTEMS ENGINEERING AND ELECTRONICS , 2000, 22(12): 49-51.

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[9] W. Li, H. Pham: An inspection-maintenance model for systems with multiple competing processes, IEEE Trans. Reliable. 2005, 54, 318-327.

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