Transcript Document
Grouping of mainteance activities
Jørn Vatn, NTNU
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Introduction to grouping
Grouping of maintenance activities is primarily done in
order to take advantage of sharing so-called set-up
costs, S, and to reduce the administrative effort when
implementing a maintenance program
Usually initial maintenance intervals for components are
established by analyzing each component separately
The easy way forward is then to group activities that have
intervals in the same order of magnitude
Optimization methods exists, and we differentiate between
Static grouping where the groups are fixed thought the entire
horizon
Dynamic grouping where the groups changes dynamically
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Single component model
A single component model will be the basis for grouping
models
Depending of failure mechanisms, possibilities to detect
failure progression, type of maintenance etc., we may in
principle establish the relation between the effective
failure rate, E(x), and the maintenance interval, x
The effective failure rate is the expected number of
failures per unit time as a function of the maintenance
interval
Typically the effective failure rate is an increasing
function of the maintenance interval
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BRP – Block replacement policy
One of the classical maintenance model is the block
replacement policy, BRP
In the BRP a single component is maintained to a good as
new condition at intervals of length x independent of
whether failures has occurred or not in the interval
Time to failure when the component is not maintained is
assumed to come from a statistical distribution, typically
the Weibull distribution
The effective failure rate as a function of x may be found
by dividing the renewal function, W(x), by the interval
length x
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Effective failure rate, Weibull situation
Since the renewal function, W(x), is numerically expensive
to find, we need an approximation formula
Chang et al (2008) has proposed the following
approximation in the Weibull situation:
𝜆𝐸 (𝑥) =
𝑊(𝑥)
𝑥
≈
Γ(1+1 𝛼 ) 𝛼
MTTF
𝑥 𝛼−1 ⋅ 𝛾(𝑥, 𝛼, MTTF
where is the shape parameter, MTTF is the mean time
to failure without maintenance, and () is a correction term:
𝛾(𝑥, 𝛼, MTTF) = 1
0.1𝛼𝑥 2
−
MTTF2
+
5
(0.09𝛼−0.2)𝑥
MTTF
Notation
cPi
Planned maintenance cost, exclusive set-up cost.
Typically the costs of replacing one unit periodically.
cUi
Unplanned costs upon a failure. These costs include
the corrective maintenance costs, safety costs,
downtime costs, and costs due to material damage.
Set-up costs, i.e., the costs of preparing the
preventive maintenance of a group of components
maintained at the same time. We assume the same
set-up costs for all activities.
E,i(x) Effective failure rate for component i when maintained
at intervals of length x. x is local time (age).
S
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Notation, cont
Mi(x)
M i ( x) x ciU E ,i ( x) = expected costs due to
failures in a period [0,x) for a component
maintained at time x, exclusive planned
maintenance cost
i(x,k) i(x,k) = [cPi + S/k + Mi(x)]/x = average costs per
unit time if x is the length of the interval between
planned maintenance, and the set-up costs are
shared by totally k activities
*i,k
The minimum value of i(x,k), i.e., minimization
over x
x*i,k
The x-value that minimizes i(x,k)
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Notation, cont
T
Planning horizon, or time between maintenance
windows
t
Running time (calendar time)
t0
Now, i.e., time of planning
ti
Time of last maintenance of component i
ki
Number of activities sharing the setup cost (dynamic)
li
How often a maintenance occation is utilized
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Static grouping
In static grouping, the groups are fixed for the entire
planning horizon
We have to stick to the grouping and the intervals even if
new insight is gained, for example related to the failure
rate
We differentiate between
Indirect grouping where the groups are formed by an indirect
approach
Direct grouping where the groups are formed explicitly one way or
another
Combination of indirect and direct
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Indirect grouping
Assume there is an occasion for preventive maintenance
every T time units
There are altogether n maintenance activities to be carried
out
cPi = individual PM cost
cUi = individual cost upon failure, CM + losses
S = setup cost
E,i(x) = effective failure rate
Each component is maintained every liT time unit
Challenge: Obtain T and li, i = 1,..,n that minimize total cost
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Indirect grouping, cont
Average cost per time unit for given T, and l
C(T,l) = S/T + i=1:n [cPi + Mi(liT)]/(liT)
= S/T + i=1:n [cPi /(liT)+ cUi E,i(liT )] (*)
The problem is a mixed continious-integer programming
problem which is a very difficult problem to solve
Proposed heuristic
1.
2.
3.
4.
Choose an initial value of T that corresponds to the smallest
individual maintenance interval i, i.e., minimizing
C(i) = (S+ cPi )/i + cUi E,i(i)
Choose li i /T
Keep li fixed and minimize (*) wrt to T
GoTo 2 and vary li slightly to check if a better solution may be
obtained
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Direct grouping
Maintenance activities are partitioned into m groups
Each group, say Gj, is a subset of {1,2,..,n}
Gj Gk = Ø, and j Gj = {1,2,..,n}
Activities in each group are maintained at the same
interval, say Tj
Cost per time unit
C(T) = j=1:m {S/Tj + i Gj [cPi /Tj + cUi E,i(Tj )] } (**)
Optimization problem
1.
2.
Given the partitioning, Gj, j = 1,..,m
… it is straight forward to minimize (**) term by term
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Direct grouping, cont
Proposed heuristic
1.
2.
3.
4.
5.
Find individual maintenance interval i, i.e., minimizing
C(i) = (S+ cPi )/i + cUi E,i(i)
Sort the intervals in increasing order, i.e., (1) < (2) < …
Look for clusters in the intervals, and let these forms groups G1,
G2,…
Given this partitioning, Gj, j = 1,..,m, minimize (**) wrt T, i.e.
C(T) = j=1:m {S/Tj + i Gj [cPi /Tj + cUi E,i(Tj )] }
GoTo 3 and vary the groups slightly to check if a better solution
may be obtained
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Dynamic grouping
In dynamic grouping the groups are not fixed
The idea is to establish the groups “on the fly”
This will enable
To update the strategy when new information becomes available,
e.g., new failure rate estimates
Reschedule the plan if opportunities arise, e.g., upon a failure
there will be an opportunity for advancing the next planned
preventive maintenance
The repair of a component failure may be postponed to the next
preventive maintenance
Take into account that the usage of a component is not fixed
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Proposed heuristic
Step 0 Step 1 Setp 2 Step 3 -
Initialization
Tentative plan
Establish the candidate groups
Optimize execution time for each candidate group,
and choose the candidate group with the lowest cost
Step 4 - Proceed with the next group, and GoTo Step 1
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Step 0 - Initialization
Let i(x,k) be the expected cost per unit time when
component i is maintained together with k-1 components
at intervals of length x:
cip S / k U
i x, k
ci E ,i x
x
The value of x that minimises i(x,k), say xi,k*, could be
found by a fix point iteration scheme (Weibull modell):
p
MTTF
c
i
i S /k
i
xi*,k
(1 1/ i ) ciU ( i 1) ( xi , i , MTTFi ) xi '( xi , i , MTTFi )
Where ’() is the derivative of the correction term:
0.2 x 0.09 0.2
'( x, , MTTF)
2
MTTF
MTTF
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Step 0 - cont
We do not know ki, hence, xi,k* and i,k*= ii(xi,k*,k) are
calculated for an initial guess of the average ki
xi* = “average” xi,k*
i* = “average” i,k*
The values of xi* and i* are calculated only once
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Step 1 - Tentative plan
Find the individual component due dates, ti* according to the
formula ti*= xi* + ti
xi *
ti
(last mtnce)
t0
ti*
T = planning horizon
(now)
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1 - Tentative plan, cont.
x4 *
x3 *
x2*
x1*
t2
t1 t4
(last mtnce)
t3
t0 t2* t1*
t4*
(now)
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t3*
T = planning horizon
2 – Candidate groups
x4 *
x3 *
x2*
Candidate group 1.a
x1*
t2
t1 t4
(last mtnce)
t3
t0 t2* t1*
t4*
t3*
(now)
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T = planning horizon
2 – Candidate groups
x4 *
x3 *
x2*
Candidate group 1.b
x1*
t2
t1 t4
(last mtnce)
t3
t0 t2* t1*
t4*
t3*
(now)
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T = planning horizon
2 – Candidate groups
As we add more
activities to the 1st
candidate group we
save set-up cost
However, there are
penalties of shifting
each individual point of execution
At some point these penalties
exceeds the savings in set-up cost
At this point we stop adding more
activities to the 1st candidate group
x4 *
x3 *
x2*
Candidate group 1.c
x1*
t2
t1 t4
(last mtnce)
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t3
t0 t2* t1*
(now)
t4*
t3*
T = planning horizon
2 – Candidate groups, cont
Note
We also have to stop searching for further candidate groups if one
activity tentatively is repeated twice within the range of the interval for
the candidate group
x4*
x4*
x3 *
x2*
x1*
t0
(last mtnce)
T = planning horizon
(now)
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3 - Optimal execution time
For a given candidate group, Kk we find the next execution
time, t*, by minimizing the following cost elements
Set-up cost
Component specific preventive maintenance cost
Deterioration cost from now to t
Average maintenance and deterioration cost from t to T
c1 (t; k ) S ciP M i (t t0 xi ) M i ( xi ) (T t )*i
iKk
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3 - Optimal candidate group size
Components not included in a candidate group are
assumed to be maintained at their average optimal
interval, yielding a cost from t0 to T:
c2 (t; k )
P
*
*
*
c
S
/
k
M
(
x
)
M
(
x
)
(
T
t
)
i
i
i
i
i
i
i
i
iKk
Total cost used to compare candidate groups is thus
c(t ; k ) S ciP M i (t t0 xi ) M i ( xi ) (T t )*i
iK k
ciP S / ki M i ( xi* ) M i ( xi ) (T ti* )*i
iKk
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4 – Next group
To proceed, we now set the clock: t* t0
x4*
x4*
x3 *
x2*
x2*
x1*
x1*
t*
t0
Group 1
T = planning horizon
t0
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4 – Next group
Then we start forming candidate group 2 and so forth
x4*
x4*
Candidate group 2.a
x3 *
x2*
x2*
x1*
x1*
t*
t0
Group 1
T = planning horizon
t0
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Comments
Wildeman (1996) proposed to see at least one group ahead when
fixing the first group (maintenance package)
We only consider the first group, and assume that subsequent activities
are executed at average optimal point of times
This simplifies the procedure significantly
In some cases this may lead to a non optimal first group
Wildeman did not consider the situation of repeated activities
In our heuristic this is rather easy
Wildeman used cPi + S instead of cPi + S/k to establish the tentative
plan
This will in some situation give different ordering of due activities, and
hence give weaker results in our opinion
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Comments, continued
Initially, the ki’s were set more or less arbitrary
After “running” the procedure, we may record the average
values of the ki’s
These values may be used to iterate for a better solution
Also note that the procedure may be improved:
When two subsequent groups have been established, we may
reduce cost by moving one activity from one of the group to the
other group
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Opportunities
The process of forming groups will in principle establish a
plan for maintenance through the entire planning horizon
T
The plan is followed if nothing special happens
If new evidence become available, or an opportunity for
reducing maintenance cost, e.g., a failure that may share
set-up cost we may advance the execution of the next
group of preventive maintenance activities
Recalculation of the grouping
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