Research Journal of Applied Sciences, Engineering and Technology 4(21): 4362-4366, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 31, 2012
Accepted: April 17, 2012
Published: November 01, 2012
Availability Modeling and Analysis of Equipment Based on Generalized
Stochastic Petri Nets
Zheng Beirong, Xie Xiaowen and Xue Wei
College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou Zhejiang
325035, China
Abstract: To meet the challenge of dramatically market, manufacturing process requires the high production
efficiency and the stability of the machining accuracy. Thus availability is the one of crucial measurements of
complex equipments. This study addresses the problem of evaluating the performance of equipment which
subject to both wear-out failures and random failures. Moreover, the different degradation states of the
equipment are analyzed and then, the model of the states is established using generalized stochastic Petri nets
and the analytical method for assessing the availability is also built up. Finally, the availabilities under different
maintenance strategies are compared and the effectiveness and efficiency of the method is verified.
Keywords: Availability, equipment, generalized stochastic petri nets, modeling
INTRODUCTION
High-performance equipments are the most important
devices in the manufacturing enterprises, such as the highprecision machine tools, AGVs. With the development of
the manufacturing technology, the machine tools are
becoming higher precision, stability and complexity.
However, by the interference from external and internal
factors during the operation, the performance of the
equipments shows a certain random, dynamic fluctuations
which cause the equipment performance degrade or even
fail (Guo et al., 2008). Thus, the production line, which is
composed of the m stages of high-precision machine
tools, meets the obstruction, cost-inefficient and low
production rates because of the key equipments
degradation or failures (Mittal et al., 2008; Sun et al.,
2012).
The availability of equipments is one of crucial
performance measures of equipments subject to failures.
The states of equipments are characterized by normal,
degradation and failure. Moreover, the failures are often
categorized wear-out failures and random ones. Most of
the researches have been carried out to discuss the
modeling of the equipment performance under one type of
failure. Vaurio (1999) presented a model to illustrate
failure, repair and maintenance behaviors and the
objectives of optimization to minimize the total cost of
two intervals. Dhouib et al. (2010) proposed an analytical
model for assessing the steady-state availability of
production lines and a model was also developed to
simulate the dynamic behavior of production lines. Wei
et al. (2011) established a dynamic availability model for
CNC system and simulated based on the Monte-Carlo.
Zio et al. (2007) presented a Monte Carlo simulation
method to model the operational dependencies with all the
complex system whole state.
Allen and Miroslaw (1998) summarized the various
models and tools in which both analytic and simulation
techniques were applied such as Markov chains, fault
trees and Petri nets. Petri nets is a powerful tool to model
Discrete Event Dynamic System (DEDS) and describe the
changes of the status of equipments. The different states
of equipments, such as normal, deterioration, failure and
the maintenance, can be represented by the places and
transitions of Petri nets. Zhang et al. (2009) built the
reliability model of a non-series manufacturing systems
with buffers, moreover, through the function reliability
index, overall evaluation of the reliability of the whole
manufacturing was carried out. These contributions haven
considered the impact of random breakdowns and wearout breakdowns on the overall performance of equipments
at the same time yet.
This research studies complex equipment subject to
both random failures and wear-out failures. Its aim is to
develop an analytical model for calculate the availability
of equipment compare with of different maintenance
strategies.
METHODOLOGY
Availability definition of equipment: For repairable
systems, availability is a more meaningful measure than
reliability to evaluate the maintenance systems since it
includes not only reliability but also maintainability
(Chiang and Chen, 2007).
Corresponding Author: Zheng Beirong, College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou
Zhejiang 325035, China
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4362-4366, 2012
Availability of a repairable system is the probability that
the system is operating during a specified time t.
Availability measures are classified by either the time
interval of interest or the mechanisms for the system
downtime. Availability of an equipment is often defined
as Eq. (1):
U = MTBFi / MTBFi + MTTRi
(1)
where, MTTRi - mean time to repair, MTBFi - mean time
between failures is the predicted elapsed time between
inherent failures of a system. In the manufacturing system
area, availability of the equipment, which reflects the
maintenance feature accurately, is more suitable to
measure the performance of equipment than reliability.
The full life cycle of equipment are divided into
working state, maintenance state and failure state,
moreover, the working state also a degradation process. A
maintenance operation makes the equipment back to
optimum state in the deterioration. Reasonable
assumptions are made in this study to simplify the model:
C
C
C
The equipment is in working state, maintenance state
or failure state.
The equipment is back to the optimum state after
maintenance operation.
The degradation level of equipment is monotonically
increasing and it can be reversed only by
maintenance behavior.
Assume the operation of the equipment is DEDS and
then availability of the equipment is defined as the
stability of the system under the working state. Suppose
1, 2, …, L are the different degradation level of the
equipment. L is the last state of equipment, that is failure
state and it can’t be back to other state anymore except
repair operation. V1, V2, …, Vi are the time of the
corresponding deterioration states. Vi is the maintenance
time of ith degradation state. So, the availability is given
by Eq. (2) (Xie et al., in press):
L −1
U=
∑V
i
i =1
(2)
V
where, V is the total time of equipment operation.
The average availability is also defined as Eq. (3)
(Xie et al., in press) since the time of different
deterioration state fit the extended time distribution:
L −1
E (U ) =
∑ E (V )
i
i =1
V
(3)
Availability model: In order to model the availability of
complex equipment and describe the stochastic processes
during the operation, the Generalized Stochastic Petri
Nets (GSPN), which is subclass of PN, is introduced in
this study.
GSPN: The GSPN is described as follows Chiola et al.
(1993):
Definition 1: GSPN (P,T; F,W, M0, 8) , where(P,T; F,W,
M0) is a P/T system.
P = {p1, p2, , …, pm}, is a finite set of places, m > 0 ;T =
{t1, t2, , …, tn}, is a finite set of transitions, n > 0, P ∪ T
… M and P1T … M; T is divided into two subsets, one is
the time transitions T1 and the another is the immediate
transition T2, T = T1 U T2 , T1 I T2 = φo
F: (S×T) ÷ N+ is the input function, which describes
directed arcs from places to transitions. Inhibitor arc is
allowed in the F.
W: (T×S) ÷ N+ is the output function, which
describes the directed arcs from transitions to places.
M0 : S ÷ N+ is the set of initial marks.
8 = { 81 = , 82, ……, 8m} is the set of average firing
rate of transitions. Each value of 8i is obtained from the
statistical analysis of simulation system or prediction
according to some requirements, Ji = 1/ 8i is called as the
average delay of transition ti
Under mark M, several transitions form a enabled
transition set H, that is:
C
If H is all timed transition, then the enable probability
of arbitrary time ti , H is: λi ∑ λk
C
If H includes certain number of immediate and timed
transitions, then only immediate transitions can be
enabled and which is selected depends on a
probability distribution function.
tk ∈H
Modeling: Assume that an equipment has L states, which
are represented by 1, 2, …, L, respectively. The
degradation process is monotonically increasing, that is,
after a period of deterioration, the system goes to state 2
from state 1. The system may goes to state 3 from state 2
under different rate and so on. The last degradation state
is wear-out failure and need to maintenance immediately.
Places represent the states and transitions represent the
behaviors. The process is showed on Fig. 1 based on
GSPN.
Besides the wear-out failure, the equipment also
suffers random failure. Both random and wear-out failure
can be repaired by maintenance behavior. So, the model
with the failures and maintenance is showed in Fig. 2.
When the equipment is in the failure state, the equipment
will return to the last state only after maintenance
behavior, such as the equipment reaches the random
failure, which represents by the place pF by t2F fires and
then after maintenance operation, which represents by the
transition tF2 , the system is repaired to state p2 again.
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4362-4366, 2012
1
2
p1
t 12
…
p2
L
N- 1
L
t 23
pL- 1
t L- 2L- 1
pL
t L- 1L
Fig. 1: GSPN model of the degradation process
t L1
t 1L
t L2
t 21
t 2L
TL- 1L
1
p1
2
t 12
…
p2
t 23
t F2
t 2F
F
t 1F
L
N-- 1
L
t L- 2L- 1
t L- 1F
t F1
pL- 1
pL
t LL- 1
t FL- 1
Fig. 2: GSPN model of the degradation and maintenance process
The availability analysis method is presented as
follows based on GSPN:
C
C
C
Build the GSPN model of the equipment and assign
exponential distribution delay to transitions.
Create the reachability graph R(m0): The firing
rate of corresponding transition of each arc in the
figure is given and then Markov chain is obtained
(the firing rate may be related to mark). All marks or
states are recorded as m0, m1, m2,…, mq!1, where, q is
the total number of states, that is q = | R(m0)|.
Analysis the Morkov chain: The stability is solved
by the Eq. (4), denoted by
∏
∏ = (π
0
)
, π1 ,..., π q :
q −1
A = 0, ∑ π i = 1
(4)
i=0
The performance evaluation is calculated based on
the stability J and the matrix of transitions fire rate A.
Assume U is the sub-set of R(m0), which mark the set
of the deterioration states 1, 2, …, N !1 in the Markov
chain. So, the availability is given by Eq. (5):
U=
∑π
mi ∈U
1
(5)
Case study: In order to adapt to rapid changes in the
market, machine tool companies develop new
equipments. For example, C160 series from INDEX can
modify own modules based on the different process plan
and form a new configuration. Therefore, the structure of
equipment is becoming more complicated and the analysis
of available is becoming more important than ever.
Consider a CNC which is used to process the boxtype parts. In this study, availability of this machine is
analyzed with different 4 states, that is, new state,
deterioration state, wear-out failure state and random
failure state. State 1 and state 2 represent the new and
deterioration respectively, moreover, the machine tool can
produce the parts until the state reach the 3 or 4. State 3
represents the wear-out failure and state 4 represents the
random failure respectively. The machine tool has to be
repaired to back to either state 1 or state 2.
Build the model of CNC: Figure 3 shows the machine
tool which includes 4 states. The meaning of places and
tokens are given by Table 1 and 2.
The stability of p1 and p2 is calculated according to
the Table 2 and then the availability is solved by Eq. (5).
The machine tool can be back to the level 1 by preventive
maintenance.
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4362-4366, 2012
Table 1: The meanings of places of Fig. 3
Place
Meanings
P1
Machine tool in new state
P2
Machine tool in deterioration state
P3
Machine tool in wear-out failure state
P4
Machine tool in random failure state
Table 2: The meanings of transitions of Fig. 3 and firing rates
Transition
Meanings
Firing rate
t1
State changes from level 1 to level
20.500
t2
State changes from level 2 to level
30.500
t3
State changes from level 1 to level
30.050
t4
State changes from level 1 to level
40.025
t5
State changes from level 2 to level
40.500
t6
State changes from level 4 to level
20.500
t7
State changes from level 4 to level
10.500
t8
State changes from level 3 to level
20.800
t9
State changes from level 3 to level
11.000
t10
State changes from level 2 to level
11.200
Table 3: The state set of Fig. 3
P1
M0
1
M1
0
M2
0
M3
0
M1
t9
t
t8
t
3
2
M3
t1
M0
t
t
10
t6
7
M2
t4
t5
Fig. 4: The state transfer diagram of Fig. 3
t9
P2
0
0
0
1
P3
0
1
0
0
P4
0
0
1
0
t8
p1
p2
p3
t2
t1
t5
t6
t4
p4
t
7
t3
Fig. 5: The model of machine tool only with repair
DISCUSSION
Fig. 3: The availability model of machine tool with prevention
maintenance and repair
NUMERICAL RESULTS
The state transfer diagram of Fig. 3 is showed in
Fig. 4 and is isomorphic to a Markov chain. Table 3
shows the state set of Fig. 3.
The stabilities of all state of the system are calculated
as follows based on Table 2 and 3:
⎧ M 0 = 0.6238
⎪ M = 0.0709
⎪ 1
⎨
.
⎪ M 2 = 01121
⎪⎩ M 3 = 01931
.
Then U = 0.8169, according to the model of
availability and Eq. (3).
The result given in Section 3.2 shows the availability
of machine tool which includes both the preventive
maintenance and repair. In order to analyze the
availability of other strategies, the model can be modified
simply. Assume the second strategy is used to the
machine tool, that the repair behavior is applied when the
machine tool in failure state, moreover, the preventive
maintenance don be applied during the machine tool lifecycle operation. So, the model is modified as Fig. 5. The
stabilities of the system are:
⎧ M 0 = 0.3412
⎪ M = 01910
.
⎪ 1
⎨
M
=
01002
.
⎪ 2
⎪⎩ M 3 = 0.3667
and the U = 0.7079.
Compared the different strategies, the availability
with prevention maintenance is 15% higher than without
prevention maintenance.
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4362-4366, 2012
In the one hand, the first strategy enhances the
availability of machine tool; moreover, the probability of
the machine tool working in better condition is 83%
higher than the second strategy. In the other hand, the
prevention maintenance adds the additional fee and
increases the burden of maintenance staff.
Therefore, it is difficult to achieve the best effect for
equipment if only preventive maintenance or availability
is considered. In the practice, the process characteristics,
production accuracy and cost of different products should
all be considered to obtain reasonable equipment
availability and low production cost.
CONCLUSION
In the study, the analysis and calculation of
availability of complex equipment are dealt with. GSPN
is adopted to model and compute and the impact
relationship of performance indicators is also analyzed:
C
C
Degradation process of complex equipment has the
characteristics randomness, non-linear and so on.
GSPN is adopted to model the operation degradation
process of equipment. The operation process is
considered as a progressive deterioration until it
finally fails. It monotonically increases, only and if
the corresponding maintenance operations can reduce
the equipment deterioration. The modeling approach
directly reflects the state changes on operation and
more truly reflects the procession of equipment
operation and maintenance.
According to given equipment operation modeling
steps, the availability model of complex equipment is
constructed and related places and transitions are
defined. Then, the steady-state probability of
equipment operation is calculated and the availability
is computed under different maintenance strategies
with detailed analysis results.
ACKNOWLEDGMENT
This study was supported in part by International
Science and Technology Cooperation Program of China
(Grant No. 2012DFG72210), Zhejiang Provincial Natural
Science Foundation of China (Grant No.Y1111147), Key
scientific and technological project of Zhejiang Province
(Grant No. 2011C14025), Key scientific and
technological project of Wenzhou (Grant No.
H20100092).
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