Impact of Few Failure Data on the Opportunistic Replacement Policy For

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Impact of few failure data on the opportunistic replacement policy for

multi-component systems

Radouane Laggoune a,, Alaa Chateauneuf b, Djamil Aissani a

a Laboratory of Modelization and Optimisation of Systems (LAMOS), University of Bejaia, Targua-Ouzemour, Bejaia 06000, Algeriab Laboratory of Mechanics and Engineering (LaMI), Polytech’Clermont-Ferrand, Campus des C ezeaux, BP 206, 63174 Aubiere cedex, France

a r t i c l e i n f o

Article history:

Received 22 May 2008Received in revised form

13 July 2009

Accepted 27 August 2009Available online 31 August 2009

Keywords:

Opportunistic maintenance

Multi-component system

Data uncertainty

Bootstrap

Monte Carlo simulations

a b s t r a c t

In continuous operating units, the production loss is often very large during the system shut down.

Their economic profitability is conditioned by the implementation of suitable maintenance policy thatcould increase the availability and reduce the operating costs. In this paper, an opportunistic

replacement policy is proposed for multi-component series system in the context of data uncertainty,

where the expected total cost per unit time is minimized under general lifetime distribution. When the

system is down, either correctively or preventively, the opportunity to replace preventively non-failed

components is considered. To deal with the problem of the small size of failure data samples, the

Bootstrap technique is applied, in order to model the uncertainties in parameter estimates. The Weibull

parameters are considered as random variables rather than just deterministic point estimates. A

solution procedure based on Monte Carlo simulations with informative search method is proposed and

applied to the optimization of preventive maintenance plan for a hydrogen compressor in an oil refinery.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The preventive maintenance (PM) is often carried out to

prevent or to slow down the deterioration processes. PM is a

scheduled downtime, usually periodical, in which a well-defined

set of tasks (e.g., inspection, replacement, cleaning, lubrication,

adjustment and alignment) are performed. In oil refining facilities,

the problems associated with part replacement are more

concerned than other routine maintenance activities such as

cleaning and lubricating, from the PM scheduling point of view.

This is because the direct costs due to part failure and

replacement are usually very high, and the impact of different

replacement intervals on the overall maintenance cost is often

very sensitive and significant, in addition to the safety require-

ments.

In series systems, the one-by-one preventive replacement of components improves the global system reliability on the account

of its availability, which would be largely penalized, because of

frequent shut downs for component replacements. For multi-

component systems (MCS), an optimal maintenance policy must

take account for interactions between the various components of

the system. The interactions are of three types [1]: economic

dependence, structural dependence and stochastic dependence,

which are defined as following:

The economic dependence concerns the influence of component

operation and maintenance actions on the overall system costs.

In other words, the system cost is not simply the sum of the

individual component costs. In this case, saving in costs or

downtime can be achieved when several components are

jointly maintained. In the present work, only economic

dependence is considered.

The structural dependence concerns components which struc-

turally form a part; therefore maintenance on failed compo-

nent implies actions on other components. For example,

replacing a part in an engine implies the disassembly and re-

assembly of other parts.

The stochastic dependence arises when the state of a componentinfluences the lifetime distributions of other components or

when components are subjected to common-cause failures.

This is often observed for redundant mechanical systems

where the degradation of a component leads to internal force

redistribution and therefore to overload other components.

The implementation of a PM policy requires a perfect knowl-

edge of the real system reliability and lifetime distribution, which

can be only obtained from a large number of failure data. In other

words, if large set of failure time observations is available, the

component and system lifetimes which are random variables can

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Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ress

Reliability Engineering and System Safety

0951-8320/$- see front matter & 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ress.2009.08.007

Corresponding author. Tel./fax: + 213 34 215188.

E-mail address: [email protected] (R. Laggoune).

Reliability Engineering and System Safety 95 (2010) 108–119

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be accurately described by Weibull distribution, for example.

However, for real-world systems, only few failure data are

available, especially for industries where the safety aspect is

important, such as oil refineries and nuclear power plants. In this

paper, we propose a PM policy for a MCS in the context of rare

data. The economic dependence between components is taken

into account by the introduction of an opportunistic policy

allowing for preventive replacement of non-failed components

during the system shut down (preventively or correctively). The

Bootstrap technique is used to circumvent the lack of data. The

methodology is illustrated by an application to a hydrogen

compressor in an oil refinery.

In Section 2, the relevant literature is reviewed, particularly

that dealing with MCS, the problem of uncertainties induced by

the rarity of failure data is also discussed and several ways to

address this problem are given. Section 3 provides the principle of

the bootstrap technique; its usefulness is also discussed. In

Section 4, the costs models are formulated for several main-

tenance policies. Section 5 gives the solution strategy, in particular

the opportunistic grouping rule and the algorithm allowing for

opportunistic grouping replacement including bootstrap esti-

mates. In Section 6, the proposed approach is illustrated by an

industrial application where the maintenance is first optimized

without considering uncertainties, and then the bootstraptechnique is integrated to examine the effect of uncertainties in

the Weibull parameters on the optimal strategy obtained above.

2. Maintenance policies for multi-component systems

For MCS, when no strong dependence exists between the

different components, such as a transport fleet constituted by

many vehicles; the traditional single-unit model developed by

Barlow and Hunter [2] can be independently applied to each unit,

in order to provide optimal replacement schedule. However, the

general case of MCS implies to take account for the interactions

between various components. The common planning approaches

used for multi-component manufacturing systems include the

group/block replacement models and the opportunistic main-

tenance models.

In the block/group maintenance policy, an entire group of

components is replaced at periodic intervals. The interval is

decided based on time, cost or both. The concept of opportunistic

maintenance comes from the fact that the cost of simultaneous

maintenance actions on various components would be less than

the sum of the total cost of individual maintenance actions.

Therefore, providing the opportunity to carry out preventive

maintenance on some components along with the replacement of failed ones, leads to very small additional cost, compared to

separate replacements. Under these conditions, the maintenance

decisions for one component depend on the states (aging) of the

other components in the system [3,4]. The economic dependence

is common in most continuous operating systems, such as oil

refineries, chemical processing facilities, mass-production manu-

facturing lines and power generators [3,5,6]. For this type of

systems, the single shut down cost is often much higher than the

cost of the components to be replaced. Therefore, there is a great

potential for cost savings by implementing suitable opportunistic

maintenance policy.

The maintenance and replacement policies of MCS are

extensively discussed in the literature [7–9]; a number of studies

have reviewed the various policies [1,10–12]. These reviews showthat most of the authors use simplified assumptions, or deal with

particular systems (special structure is often assumed), in order to

formulate the decision problem with less mathematical difficulty

[4,13,14]. From another point of view, most of the decision models

developed are based on dynamic programming or Markovian

approaches [6], which approximate continuous decision variables

by finite discrete state decision variables. These restrictions in

both maintenance policies and model formulations could affect

the optimality of the solutions because of the reduction of the

solution space. In addition, discrete state decision models are

often difficult to apply to systems with large number of

components and different failure distributions, because of the

astronomic number of combinations as the solution space

growths exponentially with the number of components. It is

Nomenclature

C c 0 corrective common cost related to the system, to be

paid at each repair upon failure

C p0 preventive common cost related to the system, to be

paid at each time the system undergoes a preventive

maintenance.

C c i specific corrective cost, to be paid at each replacement

upon failure of component i

C pi

specific preventive cost, to be paid at each preventive

replacement of component i

C c syst ; j expected total corrective cost of the whole system due

to failure of component j

C psys expected total preventive cost of the whole system

C i(ti) cost rate for component i (objective function of the

mono-component policy)

C mono(t) cost rate for the mono-group policy (objective func-

tion)

C (t,k1,y,kq) cost rate for the opportunistic multi-grouping

policy (objective function)

t basic preventive maintenance interval

(t ¼ mini ¼ 1;...;qti)ki integer multiplier of component i, defining the

periodicity of preventive replacements

ti time interval (age) between preventive replacements

of component i (ti=kit)

q number of system components

K the least common multiple of all ki

t i simulated lifetime of component i

t j time instant of failure of component j

N total number of lifetime simulations

N Y total number of parameter simulationsI F k;s;l ; I Rk;s;l

A½0; 1 binary variables indicating the states of failure

or operation, respectively

MCS multi-component system (a system constituted by

more than one component), otherwise, it is called a

component or a single-component system

G pk group of components to be preventively replaced at

the kth scheduled time instant

Ghh group of non-failed components to be opportunisti-

cally replaced at the kth interval [(k 1)t,kt]

F i(.) cumulative distribution failure (CDF) of component i

F sys(.) cumulative distribution failure of the whole system

(due to any component failure)

F sys,j(.) cumulative distribution failure of the system due to

component j

b Weibull shape parameter

Z Weibull scale parameter

R. Laggoune et al. / Reliability Engineering and System Safety 95 (2010) 108–119 109



ARTICLE IN PRESS

worth to note that when the maintenance optimization concerns

a large-scale manufacturing system, the simulation-oriented

approaches can be interesting and perform well [15].

It is to note that most of authors assume the abundance of

data. However, for real-world systems, we have generally few and

partial failure data, leading to biased estimation of the lifetime

parameters and consequently to inappropriate decisions on

maintenance policy. For this reason, the statistical uncertainties

(known also as epistemic uncertainties) should be included in theevaluation of the expected maintenance and emergency costs. In

order to address this problem and to give a useful assessment of

the uncertainties; several ways can be followed [16–19].

The Bayesian approach is useful when the experience feedback is

inexistent or rare and where experts’ opinions are available [20].

Starting from the subjective probabilities, this approach consists in

combining the experts’ opinion with statistical observations of the

operation feedback. It is worth to note that if a prior is made up on

the basis of the information coming from two experts, the accuracy of

the resulting failure rate will depend on the degree of independence

between the experts. For a simple case, it is shown that the relative

accuracy increases as the experts become less dependent [21] (i.e. the

reference to a common source decreases). Therefore, the difficulty

with Bayesian approach concerns the initial knowledge modeling.

Resampling methods create an ensemble of data sets, where each

set is replicated from the original sample. The Jack-knife algorithm,

introduced for estimating bias and standard errors, generates the new

samples by deleting one (or more) specific data points. In contrast, the

Bootstrap algorithm creates new data sets by sampling with

replacement; one or more data may be repeated more than once in

any resampled data set [22]. It is shown in [23], that the standard

Jack-knife may produce highly inconsistent estimates for the standard

error and/or other measures, in particular those describing percentile

estimate. Therefore, the Jack-knife method tends to be less suited for

the assessment of uncertainties in the practical applications,

especially for small sample size [24]. For this reason, the Bootstrap

technique has been used in the present work.

3. Bootstrap technique

Consider the procedure for building a Bootstrap estimate of

standard error of a parameter estimate y. Given a data vector

X =[ x1, x2,y, xN ]T , it can be possible to draw B independent samples,

X 1; X 2; . . . ; X B , from the original data with replacement (each X icontains N values). For each resample, an estimate of the required

parameter y

b, b=1,2,y,B is computed. The standard error of the

parameter SE ðyÞ is then estimated by the standard deviation of the

B replications:

SE ðyÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

B 1XB

b ¼ 1

ðy

b y

Þ2v uut with y

¼

1

BXB

b ¼ 1

y

b ð1Þ

Given the standard error of the parameter, the a-level

confidence limits may be constructed by assuming that each

percentile is normally distributed and computing y7 z aSE ðyÞ,

where z a is the standardized normal deviate corresponding to the

confidence level a (the Gaussian method). Alternatively, if a

Gaussian approximation is not acceptable and the value of B is

large enough, the appropriate confidence interval bounds may be

read from the list of the B estimates of the percentile ranked in

ascending order of magnitude (the percentile method). The Boot-strap percentile method represents a basic form of resampling.

More refined procedures for defining the confidence intervals,

such as the Bootstrap-t and the bias-corrected and accelerated (BCa)

methods, are available in [25].

Bootstrap method has become more practical with the general

availability of rapid computing and efficient software. Compared

to standard methods of statistical inference, this method is often

simpler, more accurate, requires fewer assumptions, and has large

applicability [26–28]. Resampling provides especially clear ad-

vantages when the assumptions of traditional parametric tests are

not met, as with small samples from non-normal distributions.

Additionally, resampling can address questions that cannot be

answered with traditional parametric or non-parametric methods,

such as comparisons of medians or ratios.

4. Maintenance model

Let us consider a system composed of a set of q components

arranged in series, the failure of any component leads to the

failure of the whole system. Let t1,t2,y,tq, be the time intervals

between preventive replacements of components 1,2,y,q, respec-

tively (Fig. 1). It is assumed that each replacement restores the

component to the ‘‘as good as new’’ condition. During the system

downtime, either for preventive or for corrective maintenance, it

is to decide if we can take the opportunity to replace preventively

some of the non-failed components. This decision should be based

of the reliability decrease and the risk undertaken if these

components fail before reaching the following scheduled

preventive time.

4.1. Cost structure

The replacement costs can be divided into two parts:

The first part is related to common system costs, especially the

production loss due to the system shut down and other fixed

costs, such as mobilizing repair crew, disassembling machine,

transportation, tools, etc. The common system cost is noted C c 0

for corrective replacement and C p0 for preventive replacement.

The second part is related to the specific characteristics of the

component to be replaced, such as spare part costs, specifictools and repair procedures. For the ith component, the specific

0 2 3 4 5 6 7 8 9 10 11 12

1

2

3

q

Component 1

Component 2

Component 3

Component q

Fig. 1. Scheduled preventive maintenance plan.

R. Laggoune et al. / Reliability Engineering and System Safety 95 (2010) 108–119110



ARTICLE IN PRESS

corrective and preventive costs are noted C c i and C p

i ,

respectively.

When a replacement is carried out, the impact on the total

system cost is given by the sum of common and specific costs. For

scheduled preventive maintenance, the system maintenance cost

is given by

C p

sys

¼ C p

0

þXiAG p

C p

i

ð2Þ

where G p is the group of components to be preventively replaced

at the scheduled maintenance time (i.e. all components which

reached the optimal age replacement as it will be detailed below).

For corrective maintenance, when the jth component fails, the

system is shut down and the failed component is replaced; the

opportunity can be taken to replace other critical components.

In this case, assuming that the preventive replacement cost and

the opportunistic one are the same, the corrective system cost is

given by

C c sys; j ¼ C c

0 þC c j þXiAGh

ia j

C pi ð3Þ

where C c sys; j is the corrective system cost due to the failure of

component j, Gh is the group of components to be replaced

preventively during this opportunity (Gh is defined according to a

deterioration-based rule, analyzing the cost/benefit balance as it

will be detailed later). According to the renewal theory and

assuming infinite horizon, the expected cost per unit time is given

by [29]

C ðT Þ ¼ limt -1

E ½C ðt Þ

E ½t ¼

Expected cost during one cycle

Expected length of a cycle ð4Þ

4.2. Models formulation for several maintenance policies

In the following cost models, the discount rate is not

considered as the focus is put on production units where themaintenance intervals are too short (i.e. few months), leading to

negligible effects on the optimal solution. However, for systems

with large maintenance intervals (i.e. several years), the dis-

counted cost should be considered, which can be easily included

in the herein models. In this case, the opportunistic replacement

may be more profitable if it occurs as late as possible, due to the

effect of discount rate. Whatever the cost is discounted or not, the

following models and ideas still remain valid without loss of

generality.

4.2.1. Age-based policy for single component

For a given replacement time t, the expected cost per unit time

is written by the sum of the expected corrective and preventive

costs divided by the expected cycle length for the component i [2]

C iðtiÞ ¼C c

i F iðtiÞ þ C pi ð1 F iðtiÞÞR ti

0 ð1 F iðt ÞÞdt ð5Þ

where ti is the time (age) for the preventive replacement of

component i and F i(.) its cumulative distribution function (CDF).

4.2.2. Equivalent mono-component approach

Here, we assume that all the system components are jointly

replaced, either at system failure or after a certain time t,

whichever occurs first. Knowing that any component failure leads

to system failure, it is assumed that after each replacement the

system becomes as good as new, the replacement time is

negligible compared to cycle length and the components are

stochastically independent (in fact, while the replacement time is

neglected in the cycle length, it should be considered in the

preventive and corrective costs due to extremely high production

losses per unit time). The system renewal cycle is given byR t0

Qqi ¼ 1 ½1 F iðt Þdt and the system total cost per unit time is

given by

C monoðtÞ ¼ðC c

0 þPq

i ¼ 1 C c i ÞF sysðtÞ þðC p0 þ

Pqi ¼ 1 C pi Þ½1 F sysðtÞ

R t

0 Pqi ¼ 1 ½1 F iðt Þdt

ð6Þ

This strategy seems to be suitable only for systems composedby identical components (with similar lifetime distributions).

However, when the component lifetimes are different, a waste of

money is observed when replacing reliable components under the

conditions enforced by other less reliable ones.

4.2.3. Opportunistic multi-grouping approach

For general engineering systems, the failure rates are very

different from one component to another, and hence, a more

realistic cost model has to be considered. The idea proposed in

this work lies on the optimal definition of the preventive

replacement times (ages) and grouping. In other words, the

replacement time t is selected, such as: (1) at each scheduled

replacement, a decision has to be made for each component to

define whether it should be preventively replaced or not; and (2)at each system failure, a decision has to be taken for each

component to see whether it should be opportunistically replaced

or left as it is until the next scheduled replacement. The final goal

is to plan regular preventive replacements where optimal

component grouping is defined (Fig. 1). In this way, the basic

maintenance timet is defined as the minimum replacement time

(relative to the weakest component): t ¼ mini ¼ 1;...;qti. In addition,

the maintenance intervals for various components are defined by:

ti=kit, where ki is an integer multiplier satisfying kiZ1 for

i=1,2,y,q (Fig. 1). In this context, the decision variables are

(t,k1,k2,y,kq), where t is a continuous variable and ki are discrete

variables.

The expression of the total cost per unit time requires the

consideration of the costs involved along a cycle. In this case, the

renewal system cycle is given by the expected time span between

the simultaneous replacement of all the components K t, where

K = lcm{k1,k2,y,kq} (lcm being the operator of least common

multiple). At the beginning of each new cycle, the system is

totally in the state of as good as new. As illustrated in Fig. 1, the life

cycle of the system involves several replacements of the

components, and therefore, decisions concerning opportunistic

and preventive replacements should be included in the cost

model. At each replacement time t k=kt (with k=1,2,y,K ), the

expected cost is calculated by including corrective, preventive and

opportunistic terms. This formulation allows us to write the total

expected cost per unit time as following:

C ðt; k1; k2; . . . ; kqÞ

¼

PK k ¼ 1½

Pqi ¼ 1ðC c 0 þ C c j þ

PiAGhk

C pi ÞF sys; jðktÞ þðC p0 þP

iAG pkC pi Þð1 F sysðktÞÞ

K t

ð7Þ

where F sys,j(.) is the CDF of system failure due to the jth

component. For a series system, the failure of any component

leads to the system failure, then F sys,j(.)=F j(.). Ghk is the group of

components to be replaced opportunistically when a failure

occurs in the interval [(k 1)t, kt] and G pk is the group of

components to be replaced preventively at instant kt: k/ki= Integer

for all i (i =1,y,q; k =1,y,k).

The minimization of the expected cost per unit time aims at

finding the best set of the maintenance times ti:

Find : t; k1; k2; . . . ; kq




ARTICLE IN PRESS

that minimize : C ðt; k1; k2; . . . ; kqÞ

subject to : tZ0; kiZ1 and ki are integers ð8Þ

When the number of failure data is small, there are large

uncertainties related to the dispersion of the parameter estimates.

In this case, the point estimate cannot be accepted and the

parameters should be modeled by random variables Y, in order to

include their uncertainties. The joint density function f (Y) can be

identified by statistical tools, especially the Bootstrap techniqueused in the present work. Therefore, the cost function becomes

conditional, regarding the realization of the uncertain parameters;

Eq. (7) takes the form

C ðt; k1; . . . ; kqjYÞ

¼

PK k ¼ 1 ½

Pqi ¼ 1 ðC c

0 þ C c j þP

iAGhk

C pi ÞF sys; jðktjYÞþ ðC p0 þ

PiAG pk

C pi Þð1 F sysðktjYÞÞ

K t

ð9Þ

where F sys,j(ktjY) and F sys(ktjY) are the conditional CDF, which

are defined for a specific realization of the parameter vector Y. To

compute the total expected cost, it is necessary to integrate the

conditional cost function over the probability domain of the

parameters, leading to the following optimization problem:Find : t; k1; k2; . . . ; kq

that minimize : C ðt; k1; k2; . . . ; kqÞ ¼

Z C ðt; k1; k2; :::; kqjYÞ f ðYÞ dY

subject to : tZ0; kiZ1 and ki are integers ð10Þ

This problem is more general than Eq. (8), as it considers the

parameter distribution, according to the available number of

failure data.

5. Solution strategy

As the cost functions in (8) and (10) contain continuous and

discrete variables, a solution strategy has to be developed for

efficient computation of the optimal replacement plan, especially

for large number of components. Although techniques like

simulated annealing and genetic algorithms can be applied for

general purpose solution procedures ensuring global convergence,

the number of required runs is usually high. Knowing that, in each

run (i.e. each evaluation of the expected cost), we have two nested

Monte Carlo simulation loops (i.e. outer loop for sampling the

Weibull parameters and inner loop for sampling the maintenance

cycle scenario), the reduction of the number of runs

becomes mandatory for efficient solution, even though global

minimum cannot be fully guaranteed. This is the reason why

specific procedure has been developed in this work. While the

replacement time can be easily optimized by classical algorithms,

the implication of the discrete variables ki leads to a very large

number of possible combinations. It is thus necessary to reduce

the number of considered combinations for practical systems,

without discarding the potentially optimal combinations.

The proposed solution is a search method based on conditional

information concerning the reliability levels of the components.

As the search algorithm requires a starting point, the initialsolution can be defined by optimal times for the individual

components (minimum of costs in Eq. (5)). This solution gives

reasonable initial values for ki, defined by the ratio between the

component optimal replacement time t0i and the minimum

optimal time t0min, which is written: k0

i ¼ Integer ðt0i =t

0minÞ. Due to

economic dependence, the search range for optimal groups,

defined by ki, can vary from the initial groups, defined by k0i , in

the range 71 defined by: k0i 1rkirk0

i þ1 with kiZ1. This

gives convenient bounds for optimal search and reduces strongly

the number of combinations to only three times the number of

components (instead of the factorial). The large size of random

samplings can be chosen to achieve statistically stable results.

5.1. Opportunistic grouping rule

The deterioration-based decision can be included by analyzing

the cost/benefit balance of the component to be preventively

replaced. Let us consider the case where the jth component fails at

the time t j between two scheduled maintenance times kt and

(k+1)t, as illustrated in Fig. 2. The opportunity of replacing the

component i leads to an expected cost C pi Riðt jÞ, where Ri(.)=1F i(.)

is the component reliability. If the ith component is left without

replacement, two cases are possible: either it remains operating

until the following scheduled replacement, which will cost

C pi Riððk þ1Þtjt jÞ, or it fails before, leading to system breakdown,

the corresponding cost is ðC c 0 þC c

i ÞF iðt Þ with a maximum value at

(k+1)t. The decision-making criterion for opportunistic

replacement can thus be defined by comparing the two costs. If the opportunistic replacement cost is less than the corrective one,

it is better to change the component; otherwise it can be left till

the following planned replacement. This rule is written

If : C pi ðRiðt jÞ Riððk þ1Þtjt jÞÞ

oðC c 0 þC c

i ÞF iððk þ1Þtjt jÞ

) Then, make opportunistic

replacement of component i

at the time t j.

Otherwise ) Leave it as it is till the next

scheduled replacement

It is to be noted that the probability and reliability functions, in

the above expression, are computed for a given set of the Weibull

Component i

Component j

0 τ ( k+1)τ

Failure and corrective replacement of C i

Opportunistic

replacement of C j

Cost = C j p R j(ti )

Scheduled replacement of C j

Cost = (C 0

p+C j p) R

j((k+1) )

Last

replacement of C j

Failure of C j

before (k+1 )τ

Cost = (C 0 c+ C

j c) F

j(t

i )

... k

Fig. 2. Decision cost basis for opportunistic replacement.




ARTICLE IN PRESS

parameters. During Monte Carlo simulations, the decision

changes, depending on the parameter realizations.

5.2. Solution procedure

The flowchart of the developed algorithm is depicted in Fig. 3.

After introducing the component failure data (probability

distributions and parameters), the cost parameters ðC c 0; C p0 ; C c

i ; C c i Þ

and the initial grouping configuration k0i ; the solution procedure

is given as following:

1. Perform a Bootstrap analysis on the failure data, to determinethe joint distribution of Weibull parameters f i(bi,Zi) for the ith

component. The joint parameter distribution can be built from

the marginal probability distributions of shape and scale

parameters, and the statistical correlation between them. It is

to note that any probability model fitting the Bootstrap output

can be used for the marginal distributions of the parameters.

2. Generate a random sample of the parameters bi and Zi,according to the above joint density functions.

3. Generate random samples of component lifetimes t i, accord-

ing to the failure probability distributions. The system failure

time is defined by: t sys ¼ mini ¼ 1;qt i and the corresponding

failed component producing the system shut down is

identified. The replacements are scheduled at the times kt,

where k is an integer varying from 1 to K .

4. At the kth replacement, the simulated system failure time t sysis compared to the scheduled time for preventive main-

tenance kt. Two possibilities exist:

(a) If no failure is observed before kt, the preventive

maintenance can be carried out at kt, according to

the current grouping rule, as defined in the updated

plan, and a move to the next scheduled time (k+1)t is

done.

(b) If failure is observed, the system is down and the failed

component is correctively replaced. On the basis of the

conditional strategy described above, the opportunity of

replacing other components is considered and the related

preventive costs are computed.5. For the replaced components in step 4, new lifetimes are

generated (as new components are installed); a move to the

next replacement time is performed (k+1)t, and so on, until

the end of the system cycle (until the replacement of all

components simultaneously).

6. For the simulated scenario, the life cycle length and the

corresponding total cost are computed.

7. Repeat steps 2–6, to generate new scenarios by random

sampling, until the prescribed number of simulations is

reached.

8. The expected total cost per unit time is estimated in terms

of the mathematical average of the computed costs and

the cycle span of the simulated scenarios. For all the

sampled scenarios, the total cost in terms of the statistical

Define the replacement plan

Generate component

lifetimes t i k

Is failure observed before the next maintenance

time: ti k < t m ?

Yes

Stop production: C 0 c

Replace the failed component: C r c

Check the opportunity

of preventive replacements: SC i p

Regenerate

new lifetimes

for replaced

components

Is the end

of the life cycle

reached ?

Yes

Conditional preventive

replacement of critical

components: SC i p

No

No

Compute the expected total cost

of the life cycle and search for

the minimum cost

Modify the maintenance plan toward the optimimum

Perform Bootstrap analysis

on available failure data

Sample the Weibull parameters

R e p e a t u n t i l t h e e n d o f p a r a m e t e r s i m u l a t i o n s

Fig. 3. Flowchart of the solution algorithm.




ARTICLE IN PRESS

expectations is given by

E ½C ðt Þ ¼ 1

N Y

XN Y

l ¼ 1

1

N

XN

s ¼ 1

XK

k ¼ 1

C c 0 þ C c

i þX

iAGh

C pi

0@

1AI F k;s;l

0@

0@

þ C p

0 þX

iAG p

C p

i0@ 1AI R

k;s;l1AÞ

ð11Þ

where N Y is the total number of parameter simulations, N is

the total number of lifetime simulations, I F k;s;l and I Rk;s;l

are

binary indicators for the states of failure and operation,

respectively. They depend on the replacement interval k, the

lifetime sample s and the parameter sample l. For the kth

replacement interval, these indicators are defined by

I F k;s;l ¼ 1; I Rk;s;l

¼ 0 if failure

I F k;s;l ¼ 0; I Rk;s;l

¼ 1 if operation

(

9. The procedure is repeated for different replacement intervals

t and grouping configurations ki. The search for theoptimum solution allows us to update the scheduled

maintenance plan, by changing t and k i, in order to define a

better combination of component grouping. The iterative

scheme is stopped when the optimum solution cannot be

improved.

The large number of random simulations ensures the stability of

the cost estimate and guarantees the solution convergence to the

optimal plan. Several numerical tests and applications have

shown that the expected cost estimate becomes stable for more

than 500 Monte Carlo samples. High precision can be reached for

cost estimate with 10,000 samples, where the coefficient of

variation is always less than 0.1%; this number of Monte Carlo

simulations is considered in the presented numerical applications.

As all significant combinations are considered, the optimal

solution cannot be missed in the proposed procedure.

Fig. 4 gives an illustrative example for a sample of five

components, where the scheduled preventive replacement has a

span equal to t. The above described algorithm is applied to

generate random scenarios. The component lifetimes are

firstly generated, as shown in Fig. 4a; it is observed that

component 2 fails before the scheduled replacement and should

be correctively replaced. Given the deterioration of component 1,

the opportunity of replacing it preventively is carried out, as

illustrated in Fig. 4b. New lifetimes are then generated for the new

components (i.e. components 1 and 2). The scheduled

replacement at t is now examined; in this example, components

1–3 are preventively replaced (Fig. 4b). In the next replacement

interval (tot r2t), components 1 and 3 appear to have high

deterioration rate and are then replaced at 2t (Fig. 4c). Then,

component 5 shows a failure before reaching the scheduled

replacement at 3t, and so on. This illustration shows how a

component can be dynamically considered in function of the

possible opportunities.

6. Industrial application

The proposed methodology is applied to a centrifugal com-

pressor, located at Skikda refinery, which is the most important oil

refinery in Algeria and among the most important in Africa. The

multiple staging compressor is driven by a steam turbine; it is

essentially constituted by the stator (diaphragms, landings,

tightness subsystem) and the rotor (shaft, wheels, equilibrium

piston, etc.). The compressor aims to recycle the necessary

hydrogen for the different catalytic reforming reactions, it also

participates to the catalyst regeneration. In addition it is necessary

for the unit pre-heating during operation starting after shut down.

0 t 2 t 3 t

Component 1

Component 2

Component 3

Component 4

Component 5

a

b

c

Component 1

Component 2

Component 3

Component 4

Component 5

Component 1

Component 2

Component 3

Component 4

Component 5

Failure

Corrective replacementOpportunistic replacement

Scheduled

preventive replacement

Fig. 4. Example of corrective/opportunistic/preventive maintenance simulation.




ARTICLE IN PRESS

6.1. Failure and cost data

For the five components to be considered, Table 1 indicates the

sample sizes and the observed number of failures, ranging from 8

to 15 records. Having the failure and cost data collected in this

study for the oil refinery, the two-parameter Weibull model is

fitted by the maximum likelihood method applied on the

observed failure times; the shape and scale parameters are

depicted in Table 1.The failure distributions of these components are depicted in

Fig. 5. It can be observed that the components C286 and C285

have comparable distributions, which is also the case for the

components C230 and C260.

Table 2 gives the corrective and preventive costs for different

components, as well as the production losses due to system

downtime; the large ratio of corrective to preventive costs can be

easily observed in these data. As the production losses are related

to the time necessary to carry out the replacement, it is assumed

to be practically independent of the component itself. The

maintenance optimization is carried out according to different

assumptions in order to compare the optimal solutions and to

show the benefits of the proposed maintenance plan.

6.2. Maintenance policies

First, we analyze various replacement policies on the basis of

point estimation of Weibull parameters (i.e. no uncertainty is

considered for the two parameters), in order to select the

appropriate one.

6.2.1. Single-component policy

As a first step, the policy based on separate components isconsidered for comparison purpose. The expected cost is com-

puted for each component, independently, and a one-by-one

optimization is applied (Eq. (5)).

Table 3 gives the optimal solutions for the system components.

While component C286 has the lowest optimal replacement time,

it is shown that two other groups can be possible: C285/C275 and

C230/C260, where the replacement times are close. As discussed

above, the ratios of individual optimal times can be used as a

starting point for group maintenance plan. From Table 4, these

ratios are given by: ki=ti/t1, leading to: k1=l, k2=l.4, k3=l.3, k4=3.7

and k5=3.5 (latter, the nearest integer will be used for k i).

6.2.2. Equivalent mono-component replacement policy

Instead of replacing separately the different components, onemay suggest to make simultaneous replacement of the five

components, in order to reduce the down time of the system. In

this case, the associated total cost is given by Eq. (6). The expected

cost, depicted in Fig. 6, shows a minimum at 23 days, with an

expected optimal cost of 438.16 h for the whole system. Naturally,

this solution is not optimal as it does not take into account the

specific costs related to the different components.

6.2.3. Opportunistic multi-grouping optimization

The proposed solution is based on optimal grouping of

maintenance operations (Eq. (7)). Table 4 gives the optimal

solutions for different grouping policies. It is shown that the

minimum cost is achieved when two groups are considered, for

which k1=k2=k3=1 and k4=k5=4. A group of high failure ratecomponents (C286, C285 and C275) with periodic replacement

every 27 days and a group of low failure rate components (C230

and C260) with periodic replacement every 108 days. The

expected cost is only 123.75h/day, which represents 72% of

reduction with respect to the case of equivalent mono-component

policy. This reduction shows clearly the importance of the choice

of the strategy to be applied and the interest of the proposed

model.

6.3. Effect of data uncertainties on the maintenance policies

In this subsection, we consider the uncertainties related to

Weibull parameters, due to the small size of the failure records.

These uncertainties are then included in the replacement policy inorder to underline the effect of data size and the scatter on the

optimality of the maintenance policy. It is to note that for the MCS,

we consider the opportunistic policy for the analysis, as it is the

Table 2

Production loss and maintenance costs.

Component Code Corrective

cost (h)

Preventive

cost (h)

Cost ratio

Corr./Prev.

Production losses 35,000.00 400.00 87.5

Sheathing C286 11,281.84 263.89 42.8

Sheathing C285 30,390.16 143.83 211.3

Tightness C275 33,244.00 339.95 97.8

Stub bearing C230 43,542.64 427.71 101.8

Tightness ring C260 51,856.00 955.35 54.3

Table 1

Failure data and Weibull parameters of the system components.

Code C ompon ent S ample

size

Observed

failures

Shape

parameter b

Scale

parameter Z

C286 Sheathing 24 14 1.73 486

C285 Sheathing 23 15 1.88 507

C275 Tightness 21 15 2.43 286

C230 Stub bearing 21 8 2.53 898

C260 Tightness

ring

34 14 2.14 905

Failure distribution

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0

time (days)

P r o b a b i l i t y d e n s i t y C286

C285C275C230C260

200 400 600 800 1000

Fig. 5. Probability density functions of the system components.

Table 3

Optimal solutions for individual components without uncertainties.

i Component MTBF (days) Optimal time t0i (days) Cost (h/day)

1 Sheathing C286 483 29.8 52.97

2 Sheathing C285 475 42.7 27.25

3 Tightness C275 240 38.5 32.66

4 stub bearing C230 787 126.1 10.86

5 Tightness ring C260 844 122.9 20.75




ARTICLE IN PRESS

most suitable strategy for our case (i.e. continuous production

systems).

6.3.1. Uncertainties in Weibull parameters

Having small sample sizes, it is more reasonable to use

resampling techniques for better uncertainty assessment. The

Bootstrap technique is applied to provide the statistical distribu-

tions of the two Weibull parameters: b and Z, and consequently to

estimate the means and the standard deviations of these

parameters. The Bootstrap is applied by repeating 5000 times

the resampling from the original data, each sample has the same

size as the original data set. For each generated sample, the two

parameters are computed by maximum likelihood estimates.

Then for the 5000 samples, statistical analysis is carried out for

the Weibull parameters in order to determine the probability

distributions, the means, the standard deviations and the

correlation coefficients.

As an example, Fig. 7 gives the Bootstrap distributions of the

Weibull parameters for the component C286. It can be observed

that while the scale parameter seems to be normally distributed,

the shape parameter is strongly skewed and can be represented bylognormal distribution. The scatter of the parameter estimate is

clearly observed in this figure, which points out the meaningless

of the point estimate on the basis of few failure data. The same

observations are verified for the other components of the system.

The results of the Bootstrap estimates for all system compo-

nents are depicted in Table 5. We can see that the standard

deviation increases largely when the sample size decreases, which

is particularly the case of component C230 with only 8 failure

records. Depending on the collected failure data, the statistical

correlation between the two Weibull parameters is globally small,

except for component C285. This is due to the large scatter of

failure times, where no clear trend can be observed between the

two parameters (especially with low number of data). When the

scatter of the failure times decreases, the correlation becomes

rather negative and tends to be stronger, this is particularly the

case of component C285.

6.3.2. Impact of uncertainties on single-component policy

In order to investigate the effect of parameter uncertainties on

the optimal times, we have performed random sampling of the

Weibull parameters, where each realization leads to a specific

replacement time and cost. The statistical analysis of the 10,000

random samples allows us to plot the histograms for the

replacement time and cost. Fig. 8 shows the distributions of

optimal replacement times of individual components C286, C285and C275. The optimal time distribution is less skewed for the

component C275 than for other components; this is confirmed by

its small standard deviation. In Fig. 8, it can be seen the highly

extended distribution tails of the optimal replacement times,

which highlight the fact that point estimation is not appropriate

for maintenance optimization, unless for large size of failure data.

On the basis of probabilistic Weibull parameters, the mean and

standard deviation of the optimal replacement times and

corresponding costs for individual components are given in

Table 6.

We observe that, except for component C230, the means of

optimal times are larger than the deterministic times; in addition,

the standard deviations of these times strongly depends on the

scatter of failure data (it is to note that the coefficients of variationranges from 21% to 61%). The impact of this scatter on the expected

cost is much more significant; as an example, the standard

deviation of component C230 (corresponding to the smallest

sample size) is 103h/day while its mean is only 77 h/day! This

table shows clearly that the decision making on the basis of point

estimation (independently of the data size) leads to high risk of

production losses, as the optimal cost cannot be guaranteed. In

other words, due to parameter uncertainties, the real maintenance

cost may be doubled with respect to the expected one. This high

sensitivity underlines the necessity to use appropriate description

of various levels of uncertainties (i.e. aleatory and epistemic

uncertainties).

6.3.3. Impact of uncertainties on multi-group replacement policiesThe parameter data in Table 5 are now considered for

opportunistic multi-grouping maintenance, where the optimiza-

tion procedure proposed in Section 5 is applied. As an example,

the distribution of the optimal replacement time for the grouping

strategy 1-1-1-4-4 is depicted in Fig. 9. It shows the large scatter

of the optimal replacement time. This distribution provides the

design-maker with information allowing for best selecting of the

replacement age, according to operational conditions.

Table 7 gives the means and standard deviations for the

optimal time and cost, under various opportunistic groupings.

While the parameter uncertainties lead to slight increase of the

mean values for the replacement time, it shows a standard

deviation varying from 4.8 to 6.7 days, which is very significant,

compared to the deterministic times. The means of the minimum

Table 4

Optimal solutions for different replacement groups without uncertainties.

Replacement groups k1k2k3k4k5 Expected system life cycle (days) Optimal replacement times t1/t2/t3/t4/t5 (days) Minimum expected cost (h/day)

Equivalent mono-component 23 23/23/23/23/23 438.16

1-1-1-3-3 73.8 28/28/28/84/84 128.09

1-1-1-4-4 97.5 27/27/27/108/108 123.75

1-2-1-4-4 85.8 24/48/24/96/96 136.54

1-2-2-4-4 71.1 20/40/40/80/80 155.28

Equivalent macro-component

0

500

1000

1500

2000

0

Time (days)

E x p e c t e d

c o s t ( € )

50 100 150 200

Fig. 6. Equivalent mono-component replacement cost function.




ARTICLE IN PRESS

Component C286

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Scale parameter

F r é q u e n c e

Component C286

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Shape parameter

P r o b

a b i l i t y d i s t r i b u t i o n

6 0 0

5 5 0

5 0 0

4 5 0

4 0 0

3 5 0

3 0 0

2 5 0

1 . 2 5 1 .

5 1 . 7 5 2

2 . 2 5 2 .

5 2 . 7 5 3

3 . 2 5 3 .

5 3 . 7 5 4

Fig. 7. Bootstrap Weibull parameter distributions for the component C286.

Table 5

Bootstrap Weibull parameters distributions of the system components.

Component Code observed failures Shape parameter b Scale parameter Z Correlation b vs. Z

Mean Standard deviation Mean Standard deviation

Sheathing C286 14 1.73 0.329 486 59 0.02

Sheathing C285 15 1.88 0.667 507 184 0.66

Tightness C275 15 2.43 0.454 286 37 0.17

Stub bearing C230 8 2.53 2.66 898 271 0.12

Tightness ring C260 14 2.14 1.01 905 176 0.34

Component C286

0

0.02

0.04

0.06

0.08

0.1

0.12

1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8

Optimal replacement interval (days)

D i s t r i b u t i o n

Component C285

0

0.02

0.04

0.06

0.08

0.1

1 6 2 8 4 0 5 2 6 4 7 6 8 8 1 0 0

1 1 2

1 2 4

1 3 6



Component C275

0

0.02

0.04

0.06

0.08

0.1

1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8



Fig. 8. Distributions of optimal replacement intervals for individual components.




ARTICLE IN PRESS

costs are higher for the first two groups. For all cases, the standard

deviation of the minimum cost is nearly 50% of the mean value.

While for deterministic parameters, the strategy 1-1-1-4-4

leads to lower cost than for strategy 1-2-1-4-4, the consideration

of parameter uncertainties leads to almost the same mean costs

for both strategies, 135.5 and 136.1 h/day, respectively, which is

still better than for the other groupings. Although the grouping

1-1-1-4-4 is more economical in deterministic consideration, the

strategy 1-2-1-4-4 shows a lower standard deviation for the

minimum cost: 64.3 h/day instead of 72.4 h/day for 1-1-1-4-4. In

other words, the policy 1-1-1-4-4 cannot guarantee the desired

minimum cost, given the data scatter. Therefore, the strategy

1-2-1-4-4, although with very slightly higher mean cost, is much

more robust than 1-1-1-4-4, and can better fit the target cost. This

strategy offers a better decision-making tool, regarding data

uncertainties. This comparison allows us to confirm that deter-

ministic approach does not really provide the optimum result, as

the risk of missing the target can be high.

These results show the importance of considering the influence

of small size of data on the Weibull parameter scatter and

consequently on the dispersion of the optimal cost. As shown in

this example, the optimal strategy considering uncertainties may

be different from the policy based on deterministic parameters.

One important conclusion is that the optimal strategy cannot be

independent of the size of the failure data. It may change with the

amount of input data and only probabilistic parameters can lead

to robust maintenance policy.

7. Conclusion

The proposed maintenance plan is based on multi-grouping

optimization for multi-component systems. As in many systems,

the production losses are very high and the maintenance policy

intends to increase the system availability through cost reduction.

The optimal multi-grouping is based on the analysis of isolatedcomponent replacement times. The times are rearranged to allow

group replacements, and thus reducing the whole system down-

times as well as the maintenance costs. The proposed optimiza-

tion algorithm shows effective cost reduction by selecting the

optimal grouping and time interval for preventive replacement.

Regarding the small sample size, the solution in the determi-

nistic approach may miss the optimum. As a matter of fact, taking

into account the parameter uncertainties, by the introduction of

the Bootstrap technique, showed that the optima obtained by the

proposed approach are different from those obtained by determi-

nistic approach. The consideration of parameter uncertainties

allows us to improve the solution optimality; moreover it gives,

additional information about the solution deviation from the

deterministic one, leading to very useful indicators for betterdecision making. One important outcome is to underline that

optimal strategy cannot be independent of the size of the failure

data and probabilistic parameters should be considered for robust

maintenance optimization.

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Table 6

Optimal times for individual components with uncertainties.

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Tightness ring C260 122.9 125.3 26.8 39.3 42.2

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0.05

0.1

0.15

0.2

0.25

20

Optimal replacement age (days)

D i s

t r i b u t i o n

22 24 26 28 30 32 34 36 38

Fig. 9. Distribution of replacement interval (strategy 1-1-1-4-4).

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Cost (h/day) 155.28 148.6 76.85




ARTICLE IN PRESS

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