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*= ii(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 
iKk
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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 
iKk
 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 
iK k
  ciP  S / ki  M i ( xi* )  M i ( xi )  (T  ti* )*i 
iKk
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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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