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PHM.2017.8079232

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The optimization of the maintenance policy based on
the simplified system reliability model
XiaotongSun,
—ƒ‰›ƒŠƒ‘‫ כ‬, Xiaomeng Guan, Shuyang Guo
School of Reliability and Systems Engineering
Beihang University
Beijing, China
sxttbb@buaa.edu.cn
Abstract—Degradation of multi-component system exists
widely in the field of engineering. And the degradation failure of
multi-component system is due to the overall degradation of the
system to a certain extent, or the degradation of component. In this
paper, we focus on the reliability model of multi-component
degradation system, which failure caused by degradation of
component failure, was simplified and applied to the optimization
model of maintenance policy. For each component of the degraded
system, pure degradation and degradation increment can cause
irreversible damage to it. The continuous degradation process was
considered to be linear. And the degradation increment caused by
the shock was considered to be cumulative. The shock was
random, so that subject to Poisson distribution. The reliability
model of the multi-component degradation system under the
cumulative Poisson shock is complex and not easy to be applied to
the maintenance model optimization model of the multicomponent system. From that, this paper simplified the reliability
model of multi-component degradation system under the
cumulative Poisson shock. And the simplified reliability function
was used as the constraint to improve the optimization model of
multi-objective maintenance policy based on the reliability theory
so as to facilitate its application in engineering. The feasibility of
the simplified reliability model and the correctness of the
improved multi-objective maintenance policy was verified by case.
The influence of the change of the reliability constraint value on
the whole maintenance policy was analyzed.
Keywords—reliability model;cumulative Poisson distribution;
optimization; degradation system preventive maintenance strategy;
I. NOTATION
ܹ௜௝ ሺ݆ ൌ Ͳǡͳǡʹ ǥ ሻ: The performance degradation caused by
the ݆th •Š‘… on the ݅th component
‫ܭ‬௜ ሺ݅ ൌ ͳǡʹǡ͵ ǥ ሻ : The degradation threshold of the ݅ th
component.
ܶ௠ : The time of a minor maintenance aimed at ݅ th
component
‫ܥ‬௠ : The cost of a minor maintenance aimed at ݅ th
component
‫ܥ‬௣ : The cost of each preventive maintenance
‫ܥ‬௥ : The update cost of the whole system
978-1-5386-0370-3/17/$31.00 ©2017 IEEE
ܰ: The number of preventive maintenance in one update
cycle
‫ܨ‬௞ ǣ The minor maintenance times of ݅th component in ݇th
preventive maintenance cycle
ܾ: The reliability constraint
‫ݎ‬ሺ‫ݐ‬ሻǣ The failure rate of the system in the first preventive
maintenance cycle
ߠ: The growth coefficient of failure rate
ܴே ሺܶሻ : The system reliability after ܰ th preventive
maintenance cycles
‫ܥ‬ሺܰǡ ܶሻ: The average cost of maintenance
ܲሺܰǡ ܶሻ: The availability of system
II. INTRODUCTION
Most of the engineering systems degrade over time. When
the degradation reach to a certain extent, the failure occurred.
The working environment of multi-component system is
complex. The degree of degradation in components and the
ability to withstand the shock is also different. Pure degradation
and degradation increment cause damage in various parts of the
system until failure. The degradation of the system is extensive
in real life, so the optimization of the maintenance policy for
degradation system is what the scholars concerned. Some people
have studied the degradation of the system. Ross [1] studied
generalized Poisson shock models, which has greatly influenced
the later study. Poisson shock process for non-cumulative
damage was discussed by Fink [2]. Guo [3] studied the
degradation process of multiple subsystems subject to
continuous wear and random shocks. Sanling Song [4][5]
analyzed the reliability of multi-component systems subject to
soft and hard failure process. Sanling Song considered the effect
of degradation of various components on the overall degradation
of the system under the cumulative Poisson shock and
established the reliability model. But not considered the
practicality of the reliability model, so that the reliability model
can not be applied directly. Many people have studied the
optimization of the maintenance degradation system in the
detail. Zhao [6] analyzed the cost function of the system by using
the failure rate as the effect of the maintenance parameter. Love
and Guo [7] discussed the optimal preventive maintenance
policy under the failure rate constraint assuming that the failure
process of the system follows the Weibull distribution. Barlow
[8] describes a preventive maintenance policy by considering the
relationship between the maintenance cycle and the remaining
life of the device, each maintenance cycle has a different cycle.
Different maintenance policies for degraded systems have been
studied. H.Wang [9] summarized, classified and compared
various maintenance policies. J.M [10] studied the application
of gamma processes in maintenance. Guo [11] established an
optimization model of multi-objective maintenance policy, but
not considered the practical application of reliability as a
constraint. In view of the complexity of multi-component
reliability model system under Poisson shock, this paper
simplified it and made it easy to use in mathematical analysis
and engineering application. The reliability model was used to
further improve the optimization model of the existing multiobjective maintenance policy. In addition, the influence of the
reliability constraint value on the whole maintenance policy was
analyzed.
This paper is organized as follows. Section 2 introduces the
reliability model of multi-component system under cumulative
Poisson shock and the method how to simplify it. Section 3
describes the optimal model for periodic preventive
maintenance policy and improve the model to convenient to use.
An example is given in Section 4 to illustrate the correctness of
the improved optimal model and practicality of the simplified
function of reliability. In Section 4, the influence of the
reliability value as a constraint on its overall maintenance policy
is analyzed.
III.
RELIABILITY MODEL FOR MULTI-COMPONENT SYSTEM
UNDER ACCUMULATED POISSON SHOCK AND ITS
SIMPLIFICATION
A. Reliability Model
There are many systems subject to cumulative Poisson
shock, such as Micro-electro mechanical systems (MEMS). This
kind of system suffers from two degradation process, pure
degradation and degradation increment. The pure degradation
process of the multi-component system is Dlinear degradation
process, and the degradation increases linearly with time. But
occasionally it subject to external shock led to its sudden
increase in the amount of degradation, and each shock on the
various components of the amount of degradation is
accumulated. We assume that each component is under
cumulative Poisson shock, failure of any component can cause
the failure of system. Only if the degradation below the
respective thresholds can the system work properly. From Song
[5] we can get the reliability model of multi-component system
under cumulative Poisson shock as follows:
௠
ሺ‫ݐ‬ሻ ൌ σஶ
௡ୀ଴ ς௜ୀଵ ቎Ȱ ቌ
௄೔ ିቀఓഁ ௧ା௡ఓೈ ቁ
೔
೔
మ
మ
ටఙഁ ௧ మା௡ఙೈ೔
೔
ቍ቏ ή
௘௫௣ሺିఒ௧ሻሺఒ௧ሻ೙
௡Ǩ
(1)
In this model, the system consists of ݉ parts; ݊ is the
number of shocks; The number of shock occur in time ሺͲǡ ‫ݐ‬ሻ is
subject to the Poisson distribution; ܹ௜௝ ሺ݆ ൌ Ͳǡͳǡʹ ǥ ሻ is the
performance degradation caused by the ݆th shock on the ݅ th
component, ܹ௜଴ ൌ Ͳ ; The performance degradation ൛ܹ௜௝ ൟ
caused by each shock is independent and equally distributed.
The degradation of performance of the ݅th component caused by
each shock is subject to a normal distribution, which is
ଶ
൯; ‫ܭ‬௜ ሺ݅ ൌ ͳǡʹǡ͵ ǥ ሻis the degradation threshold
ܹ௜௝ ̱ܰ൫ߤௐ೔ ǡ ߪௐ
೔
of the ith component. The parameter vector ߚ௜ is subject to a
normal distribution, that is ߚ௜ ̱ܰ൫ߤఉ೔ ǡ ߪఉଶ೔ ൯ . The reliability
model has been verified by Song [5].
B. The simplification of reliability model
Since the cumulative reliability model of the multicomponent system under the cumulative Poisson shock in
Section 2 is too complicated, containing the cumulative
distribution function of normal distribution and the sum of the
infinite terms. SR the reliability function is not conducive to the
use of analysis, especially for some mathematical
transformation. In the optimization of system maintenance
policy, the problem is more prominent. So it is necessary to find
a way to simplify the model to make it more practical. In this
paper, the reliability model of multi-component system under
the cumulative Poisson shock is processed linearly and
simplified into a linear function to facilitate practical
application.
In the practice of projects, the whole reliability function is
usually not used. Only the part of the reliability function is
valuable. According to the requirement from maintenance
policy, we can focus on the part of reliability function where the
value of reliability is high or the part of the reliability function
is changing rapidly. Moreover, it is very difficult to perform
polynomial fitting on the whole reliability function when the
concavity and convexity of the reliability function curve is
complicated. It is simple and easy to operate, and it can
guarantee the high degree of fit of the reliability function curve,
when we choose the focus interval to simplify the fitting of the
reliability function curve. The concrete simplification process is
as follows:
a)
b)
c)
d)
According to the actual system reliability requirements,
the focus interval of the reliability function is selected. The
part of reliability function with high value or the part of
the reliability function with rapidly changes should be
focused on.
Observe the concavity and convexity of the curves in the
focus interval. The points that reflects the concavity and
convexity of the original reliability function curve can be
given. The number of points should be as little as possible.
Use software such as Matlab to draw the polynomial curve
through the points which can reflect the concavity and
convexity of the original reliability function curve.
Compare the fit of the polynomial curve and the original
reliability curve.
Add a point in the part of the reliability curve with poor
fit, and we can adjust the new point until the degree of fit
of the polynomial curve and the original reliability curve
satisfies the requirement.
Sometimes, it is necessary to select the overall function of
reliability. Select all the function interval can guarantee the
integrity of the reliability data. But in the actual project, we
always choose to focus on the interval of reliability function.
Select the focus interval can guarantee the simplified curve and
the original reliability function image in height fit. We always
get simplified function as follow:
ሺ‫ݐ‬ሻ ൌ ‫ ݐ‬ଶ ൅ ‫ ݐܤ‬൅ ‫ܥ‬
the failure rate with its simplified reliability function. Apply it to
the optimization model of multi-objective preventive
maintenance policy. The mathematical expression of the system
failure rate is:
(2)
”ሺ‫ݐ‬ሻ ൌ
Not only for the reliability model of multi-component under
cumulative Poisson shock. But also the reliability model of
system under the other condition can use this method to simplify
the function of the reliability model. The use of the fitted
polynomial function instead of the original reliability function
which is easy to carry out various mathematical processing.
IV.
OPTIMIZATION MODEL OF PREVENTIVE MAINTENANCE
POLICY WITH CONSTRAINT OF RELIABILITYF
The development of preventive maintenance policies can
reduce and prevent failure of the system. The optimization of
maintenance policy can give the best combination results of
preventive maintenance cycle and equipment update cycle. It
can bring the maximum profits to the enterprise. Guo [11] has
established an optimization model for periodic preventive
maintenance policies, which considered three objects: failure
rate, maintenance cost rate and availability. The model of
optimization is as follows:
‹ሺܰǡ ܶሻ ൌ
ଵିఠ
ఊమ
ቆെ
ఠ
ఊభ
൭
భషഇಿ ೅
‫ ׬‬௥ሺ௧ሻௗ௧
భషഇ బ
ಿ
భషഇ
೅
ே்ା்೘
‫ ׬‬௥ሺ௧ሻௗ௧
భషഇ బ
஼ೝ ା஼೛ ሺேିଵሻା஼೘
ே்
ே்ା்೘ή
భషഇಿ ೅
‫ ׬‬௥ሺ௧ሻௗ௧
భషഇ బ
൅ ߙቇ
൅ ߙ൱ ൅
(3)
subject to ܴே ሺܶሻ ൒ ܾǡ ܰ ൒ ͳǡ ܶ ൒ Ͳ
The time of a minor maintenance aimed at ݅th component is
ܶ௠ ; The expense of a minor maintenance aimed at ݅ th
component is ‫ܥ‬௠ . The expense of each preventive maintenance
is ‫ܥ‬௣ ; The update cost of the whole system is ‫ܥ‬௥ . The number of
preventive maintenance in one update cycle is ܰ (update the
system at the last time), and the minor maintenance times of ݅th component in ݇th preventive maintenance cycle is ‫ܨ‬௞ . The
reliability constraint is ܾ. The failure rate of the system in the
first preventive maintenance cycle is ‫ݎ‬ሺ‫ݐ‬ሻ. Once the system
after a preventive maintenance, the failure rate of growth faster
than the last time, the growth coefficient of failure rate is ߠ.
ܴே ሺܶሻ is the system reliability after ܰ preventive maintenance
cycles. Į is a positive number in the dimensionless process,
taking Į=1.
It assume that the failure rate of equipment would be reduced
to the initial value after a preventive maintenance and the growth
coefficient of failure rate ߠ will be faster than before[11]. The
failure rate of the system after the Nth preventive maintenance
cycle is ‫ݎ‬௞ ሺ‫ݐ‬ሻ ൌ ߠ ௞ିଵ ‫ݎ‬ሺ‫ݐ‬ሻሺͲ ൏ ‫ ݐ‬൑ ܶሻ . The minor
maintenance times of ݅ th component in ݇ th preventive
்
maintenance cycle is ‫ܨ‬௞ ൌ ‫׬‬଴ ‫ݎ‬௞ ሺ‫ݐ‬ሻ ݀‫ ݐ‬. It use these two
parameters to describe the recovery of system reliability
In this paper, the reliability of multi-component system
under cumulative Poisson shock is taken as the constraint. Since
the failure rate ‫ݎ‬ሺ‫ݐ‬ሻ of the multi-component system under
cumulative Poisson shock is not easy to obtain, we can replace
௙ሺ௧ሻ
ோሺ௧ሻ
ൌെ
ோ ǡ ሺ௧ሻ
(4)
ோሺ௧ሻ
Calculate the integral on the both side of the function:
்
்
න ‫ݎ‬ሺ‫ݐ‬ሻ݀‫ ݐ‬ൌ െ න
଴
଴
ܴ ǡ ሺ‫ݐ‬ሻ
݀‫ ݐ‬ൌ െ൫݈ܴ݊ሺܶሻ െ ݈ܴ݊ሺͲሻ൯
ܴሺ‫ݐ‬ሻ
ൌ െ݈ܴ݊ሺܶሻ
(5)
It can be seen that the relationship between the reliability and
the failure rate of the system is as shown in (6).
்
ܴሺܶሻ ൌ ݁‫ ݌ݔ‬ቄെ ‫׬‬଴ ‫ݎ‬ሺ‫ݐ‬ሻ݀‫ݐ‬ቅ
(6)
After ܰ th preventive maintenance cycles, the system
reliability is:
்
ܴே ሺܶሻ ൌ ݁‫ ݌ݔ‬ቄെ ‫׬‬଴ ߠ ேିଵ ‫ݎ‬ሺ‫ݐ‬ሻ݀‫ݐ‬ቅ ൌ ൫ܴሺܶሻ൯
ఏ ಿషభ
(7)
The whole cost of maintenance mainly comprised the cost of
update, the cost of a minor maintenance and the cost of
preventive maintenance. Furthermore, the average cost of
maintenance can be deduced as follow:
‫ܥ‬௥ ൅ ‫ܥ‬௣ ሺܰ െ ͳሻ ൅ ‫ܥ‬௠ σே
௞ୀଵ ‫ܨ‬௞
ே
ܰܶ ൅ ܶ௠ σ௞ୀଵ ‫ܨ‬௞
ͳ െ ߠே ்
‫ܥ‬௥ ൅ ‫ܥ‬௣ ሺܰ െ ͳሻ ൅ ‫ܥ‬௠
‫ݎ‬ሺ‫ݐ‬ሻ݀‫ݐ‬
ͳ െ ߠ ‫׬‬଴
ൌ
ே
ͳെߠ ்
‫ݎ‬ሺ‫ݐ‬ሻ݀‫ݐ‬
ܰܶ ൅ ܶ௠
ͳ െ ߠ ‫׬‬଴
ͳ െ ߠே
‫ܥ‬௥ ൅ ‫ܥ‬௣ ሺܰ െ ͳሻ െ ‫ܥ‬௠
Žሺܶሻ
ͳെߠ
ൌ
ሺͺሻ
ே
ͳെߠ
Žሺܶሻ
ܰܶ െ ܶ௠
ͳെߠ
The availability of system can be deduced as follow:
‫ܥ‬ሺܰǡ ܶሻ ൌ
ܰܶ
ͳ െ ߠே ்
ܰܶ ൅ ܶ௠ ή
‫ݎ‬ሺ‫ݐ‬ሻ݀‫ݐ‬
ͳ െ ߠ ‫׬‬଴
ܰܶ
ൌ
ሺͻሻ
ͳ െ ߠே
Žሺܶሻ
ܰܶ െ ܶ௠ ή
ͳെߠ
A multi-objective optimization can be established. Using
linear weighting method in evaluation function [11], the
optimization model of the single-objective preventive
maintenance policy is further calculated:
ܲሺܰǡ ܶሻ ൌ
‹ሺܰǡ ܶሻ ൌ
ଵିఠ
ఊమ
ቆെ
ఠ
ఊభ
൭
ே்ି்೘
ே்
భషഇಿ
ே்ି்೘ή
௟௡ோሺ்ሻ
భషഇ
subject to ൫ܴሺܶሻ൯
భషഇಿ
௟௡ோሺ்ሻ
భషഇ
భషഇಿ
஼ೝ ା஼೛ ሺேିଵሻି஼೘
ఏ ಿషభ
భషഇ
௟௡ோሺ்ሻ
൅ ߙቇ
൒ ܾǡ ܰ ൒ ͳǡ ܶ ൒ Ͳ
൅ ߙ൱ ൅
(10)
Take the simplified curve of reliability model into the
optimized model for periodic preventive maintenance policy.
We can get the function as follow:
‹ሺܰǡ ܶሻ ൌ
ଵିఠ
ఊమ
ቆെ
ఠ
ఊభ
൭
భషഇಿ
௟௡൫୅௧ మା஻௧ା஼൯
భషഇ
భషഇಿ
௟௡ሺ୅௧ మ ା஻௧ା஼ሻ
ே்ି்೘
భషഇ
஼ೝ ା஼೛ ሺேିଵሻି஼೘
ே்
భషഇಿ
௟௡ሺ୅௧ మା஻௧ା஼ሻ
ே்ି்೘ή
భషഇ
subject to ሺ‫ ݐ‬ଶ ൅ ‫ ݐܤ‬൅ ‫ܥ‬ሻఏ
ಿషభ
Part 1
Part 2
Part 3
‫ܭ‬௜
ͲǤͲͳʹͷ݀݉ଷ
ͲǤͲͲͳʹ͹݀݉ଷ
ͲǤͳͳͷ݀݉ଷ
Ⱦଵ ̱൫Ɋஒభ ǡ ɐଶஒభ ൯
Ⱦଶ ̱൫Ɋஒమ ǡ ɐଶஒమ ൯
Ⱦ୧ ̱൫Ɋஒయ ǡ ɐଶஒయ ൯
ߤఉయ ൌ ͺǤ͸ ൈ
ͳͲି଻ †݉ଷ.
ߪఉయ ൌ ͹Ǥʹ ൈ
ͳͲି଼ †݉ଷ.
ߤఉభ ൌ ͺǤͶͺʹ͵ ൈ
ͳͲି଼ †݉ଷ.
ߪఉభ ൌ ͸ǤͲͲͳ͸ ൈ
ͳͲିଽ †݉ଷ .
୧୨ ̱൫Ɋ୛భ ǡ ɐଶ୛భ ൯
ߤఉభ ൌ ͳ ൈ
ͳͲିଷ †݉ଷ .
ߪఉభ ൌ ʹ ൈ
ͳͲିସ †݉ଷ.
ߚ௜
(11)
In the actual application, the simplified reliability curve
instead of the reliability model can be used directly. And it is
easy to obtain the best combination of preventive maintenance
cycle and equipment update cycle under the constraint of
reliability. Taking reliability of system after ܰ preventive
maintenance cycles as the constraint and integrating cost of
maintenance and availability of system, the optimization model
was established in (10). By using software, we can get the threedimensional surface of the interval of maintenance, number of
preventive maintenance in one update cycle and multi-objective
optimization function:ܹሺܰǡ ܶሻ. Through the software we can
find the lowest point of the three-dimensional surface using the
steepest descent method. And the lowest point is the optimal
value of the objective function.
By changing the value of the reliability constraint condition
value b, the influence of the reliability as the constraint on the
whole maintenance policy is analyzed. We can determine the
importance of the accuracy of the reliability function curve to
the maintenance policy
THE PARAMETERS OF SYSTEM COMPONENTS
Parameters
൅ ߙ൱ ൅
൅ ߙቇ.
൒ ܾǡ ܰ ൒ ͳǡ ܶ ൒ Ͳ
TABLE I.
ܹ௜௝
ߤఉమ ൌ ͺǤͶͻ͵͸ ൈ
ͳͲିଽ †݉ଷ .
ߪఉమ ൌ ͷǤͻͲͳͳ ൈ
ͳͲିଵ଴ †݉ଷ.
୧୨ ̱൫Ɋ୛మ ǡ ɐଶ୛మ ൯
ߤఉమ ൌ ͲǤͻ ൈ
ͳͲିସ †݉ଷ .
ߪఉమ ൌ ʹǤͳ ൈ
ͳͲିହ †݉ଷ..
୧୨ ̱൫Ɋ୛య ǡ ɐଶ୛య ൯
ߤఉయ ൌ ͲǤͲͳͳ†݉ଷ.
ߪఉయ ൌ ʹǤʹ ൈ
ͳͲିଷ †݉ଷ.
ߣ ൌ ʹǤͷ ൈ ͳͲିହ
ߣ
ߠ
ߠ ൌ ͳǤͳ
߱
߱ ൌ ͲǤͺ
ܾ
ܾ ൌ ͲǤͻ͸
ܶ௠
ܶ௠ ൌ Ͷͺ݄
ߙ
ߙൌͳ
‫ܥ‬௥
‫ܥ‬௥ ൌ ͷͲͲǡͲͲͲ‫݊ܽݑݕ‬
‫ܥ‬௣
‫ܥ‬௣ ൌ ͹ͲǡͲͲͲ‫݊ܽݑݕ‬
‫ܥ‬௠
‫ܥ‬௠ ൌ ʹͲǡͲͲͲ‫݊ܽݑݕ‬
1
0.9
Where ‫ ݅ܭ‬is the degradation threshold of ݅ th component;
ܹ݆݅ is the performance degradation caused by ݆th shock on ݅th
component; ߣ is the parameter in Poisson distribution, which
indicates the random event per unit time of average incidence
rate; ߱ is the weight coefficient.
Take the data into (1), through software to obtain its
approximate function curve. We can get the picture as shown in
Fig. 1.
0.8
0.7
0.6
reliability
V. CASE STUDY
In this section, we use a case to demonstrate the correctness
that the simplified reliability model is applied to the improved
maintenance policy. A type of mechanical system consists of
three components, the type of degradation in parts 1, 2 is
wearing, in which the performance degradation of component 1
is the greater the amount of wear and the worse the braking
performance. The performance degradation of component 2 is
the greater the amount of wear and the worse the sealing
performance. The degradation of part 3 is expressed as the length
of the crack, and the crack length has an effect on the strength of
the mechanical part. The longer the crack is, the lower the
strength of the part is. Assuming that the relevant parameters in
the degenerate model of each component are estimated by the
parameters as shown in the following table, the parameters in the
table are derived from the relevant literature [4]
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
time/h
1.2
1.4
1.6
1.8
2
5
x 10
Fig. 1. Reliability curve of multi-component system under cumulative Poisson
shock
According to the requirement from maintenance policy, in
this case, we focus on the interval of the reliability model image
but not the overall reliability curve. But in other actual project,
we select all the reliability curve if it is necessary. It can be seen
in Fig. 1, the curve is smoothly in (0.4,1) and it meet to the
requirement of this case. So we can only focus on the part of
ܴ ൐ ͲǤͶ. In this interval, the value of reliability is high and the
change of the curve is rapidly. So, it is only necessary to sample
points in the range of ܴ߳ሺͲǤͶǡͳሻ. Moreover, the complexity of
the concave and convex parts of the partial function segment is
lower than that of the whole function, so that the fitting process
can be made simply. At the same time it can increase the success
rate of fit. Observe the concavity and convexity of the reliability
curve in the interval:ܴ߳ሺͲǤͶǡͳሻ. Then choose three points that
can
reflect
the
curvature
of
the
curve
:
ሺ͸ͲͲͲͲǡͲǤͶሻǡ ሺʹ͹ͲͲͲǡͲǤ͸͸͹ሻǡ ሺͲǡͳሻ. If the fit is poor, we can
add a point in the interval of reliability curve.
The graph of the fitted polynomial function and the original
reliability curve is shown in Fig. 2. It can be seen that the fitting
degree is high in ܴ߳ሺͲǤͶǡͳሻ. The graph of the fitted polynomial
function and the original reliability curve is shown in Fig. 2.
The polynomial function through these three points is a
quadratic polynomial function, shown as in (12).
ܴሺ‫ݐ‬ሻ ൌ ͹ǤͲ͹Ͳ͹ ൈ ͳͲିଵଵ ‫ ݐ‬ଶ െ ͳǤͶʹͶʹͶʹͶ ൈ ͳͲିହ ‫ ݐ‬൅ ͳ. (12)
It can be seen from Fig. 3. that when the system reliability is
not less than 0.96, the image of optimal function of the
maintenance expense rate and the system availability is concave,
and the lowest point is obtained at (4,1439). In this point the
maintenance strategy achieve the average cost rate and system
availability to integrate optimal. The optimal maintenance
policy at this time is described as: Conduct preventive
maintenance at 1439h every time, and replace the system before
the fourth preventive service arrives.
In order to analyze the influence of the constraint of
reliability on the whole maintenance policy. Reduce the
reliability constraint value, such as ܾ ൌ ͲǤͻͶ. The graph of
optimized function of the average maintenance cost rate and the
availability is shown in Fig. 4.
The fitting degree in the interval is very high, so the
simplified reliability function can be used instead of the original
curve. Then take (12) and related parameters into (11). We can
get Fig. 3. as follow:
1
original reliability function
simplified reliability function
0.9
reliability
0.8
0.7
0.6
0.5
0.4
0
1
2
3
time/h
4
5
6
4
x 10
Fig. 2. Comparison between the curve of original reliability and the curve of
simplified reliability
Fig. 3. Three-Dimensional Function Graph of Optimal Maintenance Strategy
ܾ ൌ ͲǤͻ͸
Fig. 4. Three-Dimensional Function Graph of Optimal Maintenance Strategy
ܾ ൌ ͲǤͻͶ
As can be seen from Fig. 4, ሺͷǡͳͺ͹Ͷሻis the lowest point.
Comparing with Fig. 3.and Fig. 4, we can see that reducing the
reliability constraint value will significantly extend the update
cycle of equipment and maintenance cycle of system. The
difference is obvious in these two graph. It is very important to
select the exact and reliable function curve of reliability, which
is the constraint in the optimization model of maintenance
policy.
VI. CONCLUSION
In this paper, the reliability model of multi-component
system under cumulative Poisson shock was simplified. By
selecting the focus interval of the reliability function, the high
degree of fit of reliability curve can be obtained. In addition, the
optimization model of the maintenance policy of multicomponent system was improved. Furthermore, the simplified
curve of reliability function was used as the constraint in the
model of multi-objective maintenance policy. The correctness of
the improved optimization model of the maintenance policy of
multi-component system and the convenience of the simplified
reliability curve was proved in the case. At the same time, the
influence of the reliability value as a constraint on its overall
preventive maintenance policy was analyzed with the case. This
paper provides support and help to the practical application of
optimization model of maintenance policy of multi-component
degradation system under the cumulative Poisson shock.
REFERENCES
[1]
[2]
[3]
[4]
[5]
S. M. Ross, “Generalized Poisson Shock Models,” Annals of Probability,
vol. 9, pp. 896-898, 1981.
Finkelstein MS, Zarudnij Vl, A Shock process with anon-cummulative
damage, Reliab Eng Syst Saf, 71, 103-107, 2001.
Guo S.Y. “Degradation process and lifetime evaluation of repairable and
non-repairable systems subject to random shocks,” in International
Conference on Reliability Maintainability and Safety, ICRMS’2016 , in
press.
S. Song, D. W. Coit, Q. Feng, and H. Peng, “Reliability Analysis for
Multi-Component Systems Subject to Multiple Dependent Competing
Failure Processes,” IEEE Transactions on Reliability, vol. 63, pp. 331345, 2014.
S. Song, D. W. Coit, and Q. Feng, “Reliability for systems of degrading
components with distinct component shock sets,” Reliability Engineering
& System Safety, vol. 132, pp. 115-124, 2014.
[6]
Y. X. Zhao, “On preventive maintenance policy of a critical reliability
level for system subject to degradation,” Reliability Engineering &
System Safety, vol. 79, pp. 301-308, 2003.
[7] C. E. Love and R. Guo, “Utilizing Weibull Failure Rates in Repair Limit
Analysis for Equipment Replacement/Preventive Maintenance
Decisions,” Journal of the Operational Research Society, vol. 47, pp.
1366-1376, 1996.
[8] R. E. Barlow, “Mathematical theory of reliabilit,” IEEE Transactions on
Reliability, vol. R-33, pp. 16-20, 1984.
[9] H.Wang, “A survey of maintenance policies of deteriorating systems,”
Eur. opean J.Operational Res., vol. 139, no. 3, pp. 469–489, Jun. 2002.
[10] J.M. Van Noortwijk, “A survey of the application of gamma processes
in maintenance,” Rel. Eng. Syst. Safety, vol. 94, no. 1, pp. 2–21,
Jan.2009.
[11] Guo S.Y. “Optimization Of Maintenance Strategy for Multi-Component
System Subject to Degradation Process,” in Prognostics and System
Health Management Conference. PHM-Chengdu, 2016 , in press.
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