Opportunistic Maintenance of Multi-Component Systems Under

Structure and Economic Dependencies: A Healthcare System Case Study

Abdelhamid Boujarif

1 a

, David W. Coit

2 b

, Oualid Jouini

1 c

, Zhiguo Zeng

1 d

and Robert Heidsieck

Industrial Engineering Laboratory (LGI) CentraleSup

elec, Paris-Saclay University, Gif-sur-Yvette, France

Department of Industrial and Systems Engineering, Rutgers University, U.S.A.

General Electric Healthcare, 283 Rue de la Mini

ere, 78530 Buc, France

Keywords:

Multi-Component Systems, Opportunistic Maintenance, Reliability, Economic Dependence, Structure

Dependence, Genetic Algorithm.

Abstract:

This paper presents an opportunistic maintenance model for a multi-component system. We develop a model

that considers the ages and residual values of the non-failed components and component failure time distri-

butions. We also consider the structural and economical dependency between the items by favoring grouped

over individual replacement to reduce operation costs. We use a genetic algorithm to derive the optimal op-

portunistic maintenance plan by minimizing the long-run operational cost considering both maintenance cost

and potential penalty costs due to failure in the future. The model recommends additional preventive opera-

tions in cases where the reliability does not satisfy the quality condition and to reduce the long-run operational

cost. A sensitivity analysis shows that the optimal decision is mainly affected by the logistic cost, the interest

rate, and the planning horizon. The model’s performance has been evaluated using several real case problems,

demonstrating that the proposed method is very efﬁcient.

1 INTRODUCTION

After-sale services are essential in today’s business

world (Zhang et al., 2019). However, only companies

with efﬁcient operations can proﬁt from their client

services (El Garrab et al., 2020). The key to a success-

ful service is to design maintenance activities care-

fully, as it can reduce the system’s downtime and en-

ables the desired performance.

Different Maintenance strategies are developed to

optimize the reparation process. The corrective strat-

egy (CM) performs maintenance activity only after a

system component fails, while under the preventive

strategy (PM), a maintenance plan of a system com-

ponent is made according to the operating rules and

could lead to the replacement of non-failed items in

a preventive way. However, the availability of spare

parts is the primary key to ensure a high mainte-

nance service level. Thus, their procurement may of-

https://orcid.org/0000-0003-0641-9470

https://orcid.org/0000-0002-5825-2548

https://orcid.org/0000-0002-9498-165X

https://orcid.org/0000-0003-4937-4380

ten cause critical issues, especially if they are expen-

sive and when systems require high-reliability (Pas-

cual et al., 2017). To ensure spare parts availabil-

ity, many companies adopt a closed-loop supply chain

strategy where the failed parts, called Line Replace-

able Units (LRU), are recovered from the ﬁeld at sys-

tem failure and returned to their repair centers. The

repaired LRUs are used later to replace defective ones

in other systems. Therefore, the reparation event can

be viewed as an opportunity for preventive operations

since the failed parts must be returned to the repair

center. Replacing only the defective components in

LRU may shorten the spare part’s lifetime after main-

tenance because of the aging of the non-failed com-

ponents. As a result, using this unit may increase the

probability of the system’s failure and lead to addi-

tional logistic costs and loss of clients. Thus, oppor-

tunistic maintenance is the best strategy to provide

high-quality spare parts with minimum costs in this

closed-loop supply chain.

This paper presents an opportunistic replacement

model, which seeks near-optimal decision at a deci-

sion point by explicitly considering the ages and the

remaining life of the non-failed components, compo-

158

Boujarif, A., Coit, D., Jouini, O., Zeng, Z. and Heidsieck, R.

Opportunistic Maintenance of Multi-Component Systems Under Structure and Economic Dependencies: A Healthcare System Case Study.

DOI: 10.5220/0011669800003396

In Proceedings of the 12th International Conference on Operations Research and Enterprise Systems (ICORES 2023), pages 158-166

ISBN: 978-989-758-627-9; ISSN: 2184-4372

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

nent failure time distributions, and the scale of the

economy. We use a genetic algorithm to derive the

optimal maintenance plan by minimizing the long-run

operational cost considering both maintenance and

potential penalty costs due to failure in the future. The

model recommends additional preventive operations

in cases with high failure risk. A sensitivity analysis

shows that the optimal decision is mainly affected by

the interest rate and the planning horizon. We evalu-

ated the model’s performance using several real case

problems based on real data from General electric

Healthcare MR machines.

An opportunistic replacement policy (OM) is a

particular type of preventive replacement in which

working components of complex equipment are re-

placed simultaneously with a failed one when a down-

time opportunity has been created (Haque et al.,

2003). Different criteria are developed for compo-

nents’ selection. The age replacement policy is based

on replacing parts when it has achieved their esti-

mated lifetime (Jiang and Ji, 2002). In (Wang et al.,

2021), the authors propose an imperfect opportunistic

maintenance model for two unit series system consid-

ering random repair time. When a unit fails, the other

is maintained when it meets the required age. The ob-

jective is to minimize the average maintenance cost

by optimizing the thresholds of opportunistic mainte-

nance. (Li et al., 2021) studies a maintenance strategy

for wind farms by considering degradation failures

and random incidents. An operating part will only

be replaced if it reaches its critical age. However,

these strategies become complicated if implemented

in a multi-component system, even if it maximizes the

usage of a component. As the number of components

increases, more maintenance activities are conducted,

which may disturb production. Another issue is that

components may fail before their lifetime because of

environmental and external conditions.

Different approaches are developed to consider

the dependency between components for maintenance

planning. The block replacement strategy (BRP) sug-

gests that maintenance activities are conducted on a

block or a group of components. In (Nakagawa and

Zhao, 2013), the authors compared between different

modiﬁed BRP models using renewal process where

replacements are executed at constant and random

variable times to minimize the expected cost rates.

They analyze when the random replacements would

be better than the standard BRP. In (Rebaiaia and

Ait-Kadi, 2022), the authors build a model to group

parts based on their mean remaining useful life. In

(Laggoune et al., 2009), the authors focus on main-

tenance cost and suggest that deterioration-based de-

cisions can be included to solve the cost issue. The

model considers economic dependency between com-

ponents by replacing multiple units to minimize sys-

tem downtime. The solution is found by analyzing

the cost or beneﬁt balance of the component that can

be preventively replaced during CM activities. In

(Satow et al., 2000), a model is developed to maxi-

mize the global reliability of n subsystem of k parallel

components that follow Weibull distributions while

considering age conditions and time spent for repa-

ration. Each k parallel items are replaced simulta-

neously. Nevertheless, the BRP policy increases the

wasted components due to the early replacement.

Other researchers evaluate the decision based on

the operation costs. Saranga estimates how cost-

effective opportunistic maintenance is compared to

a later grounding (Saranga, 2004). Using a genetic

algorithm, the model computes the remaining useful

life cost, the down cost, and the cost of risk for each

component individually. In (Nilsson et al., 2009),

the authors study maintenance plan for power plants.

Their results prove the impact of discount interest on

maintenance plan. However, to simplify the model,

they use a discrete-time space to calculate the cost

of failure. In (Haque et al., 2003), the authors de-

velop an optimization model to maximize the net ben-

eﬁt gained from an opportunistic replacement. The

model calculates the system’s residual life with and

without opportunistic operations. The beneﬁt is cal-

culated from savings made in replacement, residual

life, labor time, and the proﬁt of increasing lifetime.

A genetic algorithm was used to solve the model.

It may be realized from the above review that

most of the mathematical models developed so far

are either limited in scope or burdened with exces-

sive computation, especially if the part to repair con-

sists of many deteriorating components. In addition,

few models consider the time needed to disassemble

a component. In addition, because of inﬂation, the

real value of failure cost may be discounted in the

future, especially for systems with a relatively long

mean useful lifetime.

The formulation of the model is described in Sec-

tion 2. In Section 3, we present a case problem based

on real data. The description of the genetic opera-

tors and the optimization technique is also described

in Section 3. We discuss the results in Section 4. Fi-

nally, Section 5 concludes the paper and highlights

some future research.

Opportunistic Maintenance of Multi-Component Systems Under Structure and Economic Dependencies: A Healthcare System Case Study

159

2 MODELING AND PROBLEM

FORMULATION

We consider a closed-loop supply chain of spare parts

where the failed units are recovered from the client’s

system and shipped back to the repair center for repa-

ration. The unit can be repaired only by replacing the

failed items. There are two possible choices for each

functional component: replace it preventively when

another component fails or use it without preventive

replacement. Under the second decision, the item will

either fail before the other components and lead to a

system failure or survive until another fails. We can

see that the optimization problem is dynamic because

each failure occasion creates a decision point. How-

ever, as soon as a replacement is made, one life cy-

cle for a component is ended, and a new one starts

with an identical time to failure distribution. The pro-

cess continues until we ﬁnally dispose of the LRU.

Thus, unlike many models in the literature where the

opportunity for preventive replacement is undeﬁned,

we consider a one-time window to decide whether

to replace the non-failed components. This decision

should be based on the unit’s reliability after repara-

tion, the waste of the replaced components’ residual

lifetime, and the structural dependence between com-

ponents.

Structural dependence means that components

structurally form a connected set so that maintenance

of a part requires disassembling the others. A disas-

sembly sequence exists between elements in the sys-

tem, so dismantling all the preceding components in a

disassembly sequence is necessary to reach a particu-

lar element for maintenance (Dinh et al., 2020). Dis-

assembly operation may affect maintenance duration

and the degradation level of the disassembled compo-

nents.

Therefore, the objective is to select a set of com-

ponents that are easy to replace preventively, to pro-

vide repaired parts with a minimum failure risk during

the planning horizon of T years and a minimum waste

of residual lifetime. A minimum required reliability

can also be considered as a constraint quality.

Let us deﬁne the following notations:

• ζ = [1, 2, 3, ..n]: set of part components,

• Cost

: price of component c,

• M

: the average lifetime of component c,

• RV

cost

: waste weight of component c,

• LC: labor cost,

• Cost

: logistic cost (shipping cost to replace the

LRU with a new one at the client site),

• τ

: disassembling time for component c,

• a

: age of component c,

• R

(t): reliability function of component c,

• f

(t): probability density function of failure time

for component c,

• R

sys

(t;a

, a

, .., a

) = h(R

(t;a

), ..., R

(t;a

)):

reliability function of the system as a function of

components reliability,

• f

sys

(t;a

, a

, .., a

): probability density function

of system’s failure,

• T : planning horizon,

• r: interest rate,

• D = (D

i j

)

CXC

: disassembly matrix for the system,

• s

: state of component c,



1, if component c is in a failing state

0, otherwise

One of the characteristics of spare parts reparation is

that the components may have different ages with a

large variance; the fragile ones usually would have

young ages, while the robust items would be an-

cient. Therefore, estimating the unit reliability is not

straightforward. We propose to express the unit’s re-

liability R

sys

as a function of components’ reliabil-

ity and ages. For example, for a series units, the

reliability of the part can be expressed as R

sys

(t) =

c∈ζ

(t;a

We then formulate the optimization problem as

follows.

Decision Variables: we deﬁne the binary decision

variable x

for each component c, with



1, if component c is replaced preventively

0, otherwise

Constraints: A failed component must be replaced

to restore the system to its operating state. Thus, we

can not select a part with a state variable s

= 1 for

opportunistic maintenance. As a result, we deﬁne the

relation between the decision variable x

and the state

variable s

as follows.

+ s

≤ 1, ∀c ∈ ζ (1)

Objective Function: An optimal solution mini-

mizes the total cost of maintenance, denoted by TC.

The model would suggest not replacing any compo-

nent preventively when opportunistic maintenance is

inefﬁcient.

The total maintenance cost consists of four com-

ponents. The ﬁrst cost, denoted by C

, is the sum of

the newly-bought components’ price to replace the se-

lected units. It includes the corrective replacements

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

160

as well as the preventive ones and can be calculated

as given in Eq.2.

c∈ζ

+ s

) ×Cost

, (2)

The second cost, denoted by C

, penalizes for the un-

used remaining life of the items to be replaced pre-

ventively. It is given by

c∈ζ

sys

(0;a

(1 − s

), .., a

(1 − s

))

(3)

+∞

t f

(t;a

)dt.

The waste weight of a component, denoted by RV

monetizes the useful life of an item based on its pur-

chase price. Eq.3 represents the loss value of the se-

lected items since a portion of their life is lost for op-

portunistic activity.

The third component, denoted by C

, represents

the expected cost of failure during the planning hori-

zon. When a failure occurs, the logistic cost Cost

must be counted. However, to compare the future

payment to the present time, its present value must be

calculated. It means the amount of money that should

be deposited into the bank now at a speciﬁc interest

rate r to pay for an outlay C after duration T . At time

t ≤ T , the conditional probability of failure after repa-

ration can be expressed as follows:

P(T

sys

< t) =

sys

(t; a

(1 − (x

+ s

)), .., a

(1 − (x

+ s

))

sys

(0;a

(1 − (x

+ s

)), .., a

(1 − (x

+ s

))

(4)

For the replaced components correctively or oppor-

tunistically (x

+ s

= 1), their age would be restored

to zero, while it won’t change for the other compo-

nents. So for a small variation of time, this probability

can be expressed using the calculated probability den-

sity function (pdf ) of the system f

sys

and the present

value of the logistic cost is Cost

× (1 + r)

−t

. Thus,

the total present value of the expected cost of failure

during the planning horizon, C

, can be expressed as

Cost

sys

(0;a

(1 − (x

+ s

)), .., a

(1 − (x

+ s

))

sys

(t; a

(1 − (x

+ s

)), .., a

(1 − (x

+ s

))

(1 + r)

dt.

(5)

The last cost component, denoted by C

, reﬂects the

needed time to repair the part. We use an approach

developed by (Dinh et al., 2020) to calculate the dis-

assembling time for a component group. Based on the

structure connection between components, the disas-

sembly matrix D is constructed. The elements of the

matrix are binary coefﬁcients. The parameter D

i, j

= 1

if component j must be disassembled to reach compo-

nent i for maintenance. The cumulative disassembling

time of a component c, denoted by τ

, can be deﬁned

as the sum of disassembling times for all the compo-

nents on the path of disassembly (Eq.6).

k∈ζ

× D

c,k

. (6)

For a group of components, there may be some in-

tersections between the disassembly path of different

items. As a result, the disassembly duration of the in-

tersection nodes must be counted only once, even if

it appears on several ones. Eq.7 gives the total disas-

sembly time, denoted by τ

group

, of the replaced com-

ponents. We have

group

c∈ζ

+ x

) × τ

(7)

−

c∈ζ

× max(

k∈ζ

+ x

) × D

k,c

− 1, 0),

Where the ﬁrst term represents the total disassembly

duration of all replaced components when they are re-

placed separately. The second term is the time sav-

ing due to intersections among the disassembly paths.

Note that

k∈ζ

+ x

) × D

k,c

represents the total

number of components in the replaced group that have

component c on their disassembly path. In case there

is no intersection, the second part will be equal to

zero. Therefore, the total labor cost is the total repa-

ration time times the labor cost per time unit, i.e,

= 2 × LC × τ

group

. (8)

Finally, the objective function can be written as

minTC = C

. (9)

3 INDUSTRIAL CASE STUDY

We propose in this section an application of the de-

veloped model based on a real industrial case that we

describe below. First, we present the historical data

and the approach to extract the reliability functions.

Then, we explain the solution technique used to solve

the optimization problem.

GE Healthcare is the medical branch of conglom-

erate General Electric, one of the global leaders in

sales and services of medical systems, notably those

of medical imaging. Because of the criticality of

its products (medical devices) and the technological

Opportunistic Maintenance of Multi-Component Systems Under Structure and Economic Dependencies: A Healthcare System Case Study

161

characteristics of its components, GE Healthcare of-

fers a maintenance service to its customers. The ser-

vice’s main objective is to ensure its products’ relia-

bility (reducing the failure rate occurrence) while re-

ducing unavailability simultaneously. Global service

and operations (GSRO) department within the com-

pany is responsible for service parts. This depart-

ment covers parts supply chain management, ware-

house management, choice of transport, inventory

management policies throughout the network, and re-

pair strategies for defective parts.

3.1 Experiments Design

We study the impact of the developed strategy on a

critical LRU for MR machines. We consider spare

parts composed of 11 components in series. Figure 1

and Table 1 represent the physical structure between

components and the disassembling time for each one,

respectively. For example, components 4, 5, and 7

must be disassembled before component 10 is reached

for replacement.

Figure 1: System’s structure.

Table 1: Components’ dismantling time.

Component C1 C2 C3 C4 C5 C6

Disassembling

time (U.T)

3 1 1.5 0.2 2 4.5

Component C7 C8 C9 C10 C11

Disassembling

time (U.T)

9 4.5 1 1 1

GE spare parts supply chain is so particular that

it makes gathering parts’ lifetime challenging. De-

pending on forecasted demand and the stock level of

warehouses, one part can be shipped to different ware-

houses in different countries before being installed on

a system. It can also be used as a test unit on vari-

ous systems for a few hours before reaching its des-

tination. In addition, the same part can be repaired

multiple times in different centers.

Under these conditions, using shipments’ transac-

tions, we use a tracing system to reconstruct parts’

history. We build a timeline using the collected trans-

actions set linked to the part. Finally, we extract

installation and failure dates. Based on the calcu-

lated age of the spare parts and the number of re-

pairs, we estimate the age of components using the

replacements record. Component age is restored to

0 if replaced during reparation, while we accumu-

late the operating times if the component did not fail.

We construct the lifetime distribution for each com-

ponent separately by ﬁtting different known distribu-

tions and selecting the one with the minimum value

of Akaike information criterion (AIC) metric. For a

statistical model of some data with a number k of es-

timated parameters and a maximized value L of the

likelihood function, the AIC value is the following

AIC = 2k − 2ln(L). We model parts’ reliability as the

product of component reliability distributions.

3.2 Solution Approach

We use genetic algorithm to get a near-optimal solu-

tion for the opportunistic strategy. This heuristic has

been widely applied in many cases to solve nonlinear

problems. It is characterized by balancing exploita-

tion and exploration in the search space. This bal-

ance is strongly affected by strategy parameters such

as population size, maximum generation, crossover

probability, and mutation. The procedure for the so-

lution of the opportunistic replacement model is de-

scribed below:

• We select the possible solutions based on two con-

ditions :

– The failed components should not be replaced

opportunistically to satisfy constraint 1.

– Since the main objective of the company is to

minimize the early failure of the repaired units,

we deﬁne the constraint below to set the num-

ber of failures before 6 months less than 10%.

sys

(180;a

(1 − (x

+ s

)), ..)

sys

(0;a

(1 − (x

+ s

)), ..)

≥ 0.9. (10)

• The classic genetic algorithm chooses a single

ﬁxed mutation rate for all solutions, regardless of

their ﬁtness values. As a result, mutations can

disturb good and bad chromosomes equally. We

use the adaptive technique with two ﬁxed proba-

bilities p

max

= 1 for the low-quality solutions and

min

= 0.4 for the good ones. The bad solutions

have a ﬁtness value less than the population’s av-

erage, while the high-quality chromosomes are

those with a higher value.

• We terminate the algorithm evolution if the ﬁtness

does not change for 25 consecutive generations.

• To minimize the ﬁtness function computational

time, we save the evaluated individuals’ score so it

can easily be extracted if the solution has already

been evaluated in previous iterations.

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

162

4 RESULTS AND ANALYSIS

Table 2 represents the purchase price for new compo-

nents and their average useful life Mul in Unit of time

(U.T). We consider logistic cost equal to 750 Unit of

Cost (U.C) and an interest rate of 15%. To evaluate

the model, we deﬁne two metrics: the net beneﬁt and

beneﬁt ratio, to compare the CM strategy to the op-

portunistic one. The net beneﬁt, denoted by NB, is

the difference between the cost TC

of the corrective

solution (x

= 0, ∀c ∈ ζ) and the best solution, denoted

by TC

best

. We calculate the beneﬁt ratio as the saved

ratio of the total corrective cost,

Table 2: Costs parameters.

Component C1 C2 C3 C4 C5 C6

Component

cost (U.C)

22 24 6 43 140 2

Mul (U.T) 71k 22k 54k 44k 1.5k 16k

Component C7 C8 C9 C10 C11

Component

cost (U.C)

34 23 6 9 8

Mul (U.T) 68k 183k 37k 58k 24k

4.1 Computational Time

We have collected reparation data for 260 LRUs with

different ages and multiple sets of failed components.

Each part is composed of 11 components in series.

We calculate the ages of the components for all the

items. We apply the optimization model on the se-

lected parts for a planning horizon of 2 years. De-

pending on the set of possible solutions, the compu-

tational time can vary from 1 min to 30 min. Fig-

ure 2 represents the computational time distribution.

Unlike weak LRU, parts with higher strength (good

global reliability) have many possible solutions. As a

result, computing the best solutions usually takes 15

to 20 min.

Figure 2: Computational time distribution.

4.2 Impact on Parts Useful Life

In general, the developed strategy improves spare

parts reliability. Figures 3 and 4 compare the esti-

mated conditional reliability before and after conduct-

ing opportunistic replacements. They represent, re-

spectively, the gain in survival probability of 6 months

and 2 years after maintenance operations. Each data

point corresponds to a unique spare part with de-

ﬁned ages for its components. It represents the dif-

ference between part’s reliability under CM and OM

strategies. The improvement in survival probability is

larger if it is evaluated at the end of the planning hori-

zon. In fact, the model suggests opportunistic replace-

ments for all parts with low reliability. As a result,

most of the reliability after two years of the simulated

parts becomes higher than 0.5. The model also recom-

mends minimum replacements for old units to satisfy

the quality constraint. In some cases, parts reliability

can be improved by 0.5 points just by changing few

sets of components.

Figure 3: Gain in survival probability of 6 months.

Figure 4: Gain in survival probability of 2 years.

Consequently, the additional replacements help

improve the remaining lifetime of the repaired parts.

Figure 5 represents the gain in Residual life after op-

portunistic maintenance. It is the difference between

the estimated remaining useful life before and after

adopting the OM strategy. For some parts, chang-

ing some components opportunistically can add more

than one year to their lifetime.

Opportunistic Maintenance of Multi-Component Systems Under Structure and Economic Dependencies: A Healthcare System Case Study

163

Figure 5: Estimated gain in parts lifetime.

4.3 Cost Analysis

In this section, we analyze the efﬁciency of the sug-

gested replacements using the deﬁned metrics (net

beneﬁt and beneﬁt ratio). We consider 3 cases with

different component’s average age. Table 3 summa-

rizes the components’ ages and the probability of sur-

viving after the ﬁrst six months. Table 4 represents

the decisions taken for each case. A value equal to 1

means the preventive replacement of the component.

Columns TC and TC

are the total cost of the best

and corrective solutions, respectively. NB is the ben-

eﬁt deﬁned at the beginning of this section.

Based on Table 4, all the decisions taken outper-

form the corrective strategy. The net beneﬁt increases

with the risk of failure. The more the part is likely

to fail, the more consistent OM decisions are. In the

ﬁrst case, the corrective operations satisfy the deﬁned

quality constraint. However, the model chooses to

preventively change the cheapest and easiest compo-

nents to reduce the failure cost. In the second case, the

model selects linked components to replace, to maxi-

mize the part’s reliability and minimize the labor cost.

In the third case, the model recommends changing the

most critical components, C5 and C11, since their RV

is low compared to the other items.

4.4 Sensitivity Analysis

Sensitivity to the Planning Horizon T . Table 5 re-

sults lead to a twofold conclusion . First, the total cost

increases over time because of the failure risk. The

longer the horizon plan and the lower chance of sur-

vival. Thus, the model selects many components to

improve part’s reliability. In addition, increasing the

planning horizon makes the token decision more con-

sistent than the corrective approach. Second, setting a

very long horizon reduces the solution’s efﬁciency. If

the remaining useful life of many components is less

than the horizon plan, it would be better to change the

planning duration to minimize the number of replace-

ments and waste costs.

Sensitivity to the Waste Cost. To analyze the waste

cost impact on the decision, we add a coefﬁcient α to

Figure 6: Impact of waste cost on maintenance decision.

Eq.9. The new total cost is expressed now as TC =

+ α ×C

From Figure 6, we notice that the number of re-

placed components decreases with the rise of α. The

cheaper the components, the higher the number of

replacements. It can easily be understood because

cheap items make it proﬁtable to replace them preven-

tively to improve the part’s reliability. Nevertheless,

when the total costs of the components are compara-

ble with the logistic costs Cost

, the net beneﬁts of the

chosen decision may drop to less than zero.

Sensitivity to Interest Rate r. As illustrated in Fig-

ure 7, the higher the rate r, the lower the number of

replacements for a ﬁxed time horizon and coefﬁcient

α. It can be explained because a high rate means we

focus on the present time and do not care much about

the future. As a result, the solution chosen is more

conservative than the one under a lower interest rate.

Figure 7: Impact of interest rate on maintenance decision.

5 CONCLUSION

In this paper, we developed a model for opportunis-

tic maintenance selection. The model considers the

age of the components, their residual value, and the

structural dependence between various items. It also

accounts for the present value of future losses due to

failures. The model recommends additional preven-

tive operations to minimize the failure cost and when

the reliability does not satisfy the quality condition.

The decision changes depending on the logistic cost,

the interest rate, and the planning horizon. The latter

helps control the life cycle of the part. If the sys-

tem is at the end of its life cycle, replacing many

components to repair the LRU may not be efﬁcient.

Nevertheless, a high horizon plan makes the model

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Table 3: Components’ age.

Case C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 R

sys

(180)

1 727 727 727 727 0 727 727 727 727 727 727 0.91

2 0 0 0 1364 637 1364 1364 1364 1364 1364 1364 0.86

3 0 0 1649 1649 1649 1649 1649 1649 1649 1649 1649 0.75

Table 4: Replaced components preventively and net beneﬁt.

Case

(U.C)

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

(U.C)

Beneﬁt

(%)

1 7671 1 0 1 0 0 1 0 0 0 0 1 8331 660 7.92

2 9012 0 0 0 0 0 1 0 0 1 0 1 9734 722 7.41

3 8788 0 0 0 0 1 0 0 0 0 0 1 13036 4248 32.58

Table 5: Impact of planning horizon T .

T (days) 180 365 730 1095

TC (U.C) 4451 5936 8788 11735

Total replacement 3 4 2 4

NB (U.C) 665 2709 4248 3125

Beneﬁt (%) 12.98 31.34 32.58 21.02

recommend conservative decisions depending on the

corrective R

conditional

(180). The model suggests not

performing other replacements if it satisﬁes the hard

constraint. However, if the corrective conditional re-

liability is lower than 0.9, the model recommends re-

placing the cheapest components to satisfy the con-

straint even though the beneﬁt is negative. Therefore,

the net beneﬁt can be used as a metric to evaluate the

consistency of the proposed solution. On the other

hand, the computational time is still challenging. A

neural network can be built and trained to simplify

the integral computation time and improve the solu-

tions’ quality. In addition, the developed model does

not consider stochastic dependency between compo-

nents. Grouping components based on the correlation

coefﬁcient can approximate the dependency. Never-

theless, it will increase the total number of replace-

ments. Therefore, another strategy to overcome this

limitation is required.

ACKNOWLEDGMENTS

The research of Zhiguo Zeng is partially ﬁnancially

supported by ANR under grant number ANR-22-

CE10-0004 and the chaire of Risk and Resilience of

Complex Systems (Chaire EDF, Orange and SNCF).

The participation of David Coit in this research is par-

tially ﬁnanced by the international visiting grant from

Centralesup

elec, and the Bourses Jean d’Alembert

from Universit

e Paris-Saclay.

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