Integrating Reliability and Sustainability: A Multi-Objective Framework

for Opportunistic Maintenance in Closed-Loop Supply Chain

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, Stochastic Dependence.

Abstract:

Closed-loop supply chains are at the forefront of sustainable industrial practices since they promote the reuse

of products through remanufacturing, recycling, and repair operations. Within this framework, repair centers

are increasingly considered an alternative source of spare parts. This creates a need to enhance repair oper-

ations to balance the reliability of repaired parts with sustainability. This paper, developed in collaboration

with GE Healthcare, presents a multi-objective framework incorporating spare part reliability post-repair esti-

mation into opportunistic maintenance decisions. This research uses real-world data and advanced modeling

techniques to reﬁne maintenance strategies and provide a comprehensive solution that acknowledges compo-

nent interdependencies. By employing NSGA-III, the paper seeks to develop a decision-support mechanism

that recommends the proactive replacement of components, thereby enhancing the quality of repaired spare

parts.

1 INTRODUCTION

Nowadays, closed-loop supply chains (CLSCs) are

leading a paradigm shift toward embracing sustain-

able industrial practices. This transformation focuses

on giving products a second life through remanufac-

turing, recycling, and repair operations. Within this

framework, repair centers are increasingly viewed as

a second supplier of spare parts.

Maintenance protocols have conventionally used

new spare parts in critical domains, such as the med-

ical sector, to guarantee system ﬁdelity. On the other

hand, in closed-loop supply chains, repaired parts are

treated equally to new ones. This assumption can oc-

casionally fall short of stringent reliability standards.

This can lead to increased equipment failures, which

impedes the circular efﬁcacy of the supply chain.

Therefore, enhancing repair operations is neces-

sary to balance reliability and sustainability goals.

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

To this end, opportunistic maintenance emerges as

a strategic intervention. This methodology replaces

failed components and those approaching their end-

of-life to reduce the likelihood of future failures

proactively.

Nevertheless, existing opportunistic maintenance

models have often assumed components indepen-

dence. This presumption may not always align with

reality, where components deterioration is frequently

interdependent. Thus, recognizing and integrating

these dependencies is imperative for developing a ro-

bust maintenance framework that aligns with the dual

imperatives of reliability and sustainability within

closed-loop supply chains.

This paper, developed in collaboration with GE

Healthcare, proposes a framework to address this re-

search gap. We developed a multi-objective frame-

work that leverages estimating the reliability of a

spare part after repair and its use for maintenance

decisions. Our partnership with GE Healthcare en-

ables us to test and reﬁne the proposed strategies us-

ing real-world data and sophisticated modeling tech-

niques. Through a nuanced analysis that captures the

interdependencies within the spare parts supply chain,

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

Integrating Reliability and Sustainability: A Multi-Objective Framework for Opportunistic Maintenance in Closed-Loop Supply Chain.

DOI: 10.5220/0012487400003639

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 179-189

ISBN: 978-989-758-681-1; ISSN: 2184-4372

179

we aim to provide a comprehensive solution to this

problem.

This research aims to develop a decision-support

mechanism that can enhance the quality of repaired

spare parts by recommending proactive replacement

of components. The study employs NSGA-III to

achieve this, which helps balance the various ob-

jectives of ensuring component longevity, maintain-

ing spare parts reliability, and achieving sustainabil-

ity goals. The ultimate goal of this research is to con-

tribute a unique perspective to the literature and estab-

lish a framework for maintenance strategies that can

be implemented across various industries to improve

both sustainability and reliability.

2 LITERATURE REVIEW

This work contributes to the body of knowledge in

three principal areas: the efﬁcacy of Closed-Loop

Supply Chains (CLSCs), the strategic implementation

of opportunistic maintenance (OM), and the nuanced

reliability modeling of multi-component systems con-

sidering inter-component dependencies. The follow-

ing subsections explore these research streams, high-

lighting the intersectionality of these domains as they

pertain to this study’s objectives.

2.1 Closed-Loop Supply Chain

Efﬁciency

The concept of CLSCs originated in the 1990s, marks

a signiﬁcant paradigm shift towards sustainable oper-

ations (van Nunen, 1995; Fleischmann et al., 1997;

Guide and Wassenhove, 2003). It represents the the-

oretical foundation for much of the recent discussion

on the circular economy (Agrawal et al., 2019). The

recovery processes integral to CLSCs, including re-

pair, remanufacturing, and recycling, have been es-

tablished as key drivers in reducing the demand for

virgin materials, curtailing energy use, and limiting

waste, thereby contributing to both environmental and

economic goals (Chen et al., 2021).

Strategic and tactical considerations, such as net-

work design and inventory management, dominate

the literature, highlighting the centrality of these fac-

tors in optimizing CLSC performance (Souza, 2013).

Within this strategic framework, (Gobbi, 2011) exam-

ined the inﬂuence of product residual value (PRV)

on recovery options, advocating for nuanced deci-

sions based on the residual value of returned products.

Scenario-based models for spare parts supply, such

as those developed by (Esmaeili et al., 2021), offer

a methodological approach to redesigning sales and

after-sales services within the CLSC, offering various

supply options. The challenges of uncertainty in sus-

tainable network design were tackled by (Guo et al.,

2022), who considered the return rates of ﬁrst and

second-hand products. The valuation of remanufac-

tured products, assuming parity in quality with new

products, was explored by (Zhang et al., 2021b). Fur-

thermore, (Tahirov et al., 2016) presented a mathe-

matical model to compare production, remanufactur-

ing, and their combination within the CLSC frame-

work.

Despite the extensive research on CLSCs, there

remains a paucity of literature addressing the quan-

tiﬁcation of repaired part quality and its consequen-

tial impact on the efﬁciency of the CLSC. This study

endeavors to ﬁll this void, proposing a quantitative

model that factors in the reliability of repaired parts

to enhance the efﬁcacy of the CLSC.

2.2 Opportunistic Maintenance

Opportunistic maintenance (OM) is recognized for

its preventive approach, which extends maintenance

activities beyond failed components to include those

at risk during opportunistic downtime (Haque et al.,

2003). A comprehensive review of OM strategies by

(Ab-Samat and Kamaruddin, 2014) covers literature

up to 2014, while (Diallo et al., 2017) focused on

maintenance strategies for second-hand products. The

literature identiﬁes age-based and block replacement

as two prevalent OM policies. Age-based policies, as

outlined by (Jiang and Ji, 2002), advocate for replace-

ment based on the estimated lifetime of components.

However, few studies address age-replacement in

multi-component systems. The work of (Wang et al.,

2021) introduced an age-based OM model account-

ing for random repair times, while (Li et al., 2021)

proposed a maintenance strategy for wind farms that

considers both degradation failures and stochastic in-

cidents.

Block replacement policies are predicated on the

replacement of components in groups, based on var-

ious criteria such as remaining useful life and main-

tenance requirements (Rebaiaia and Ait-Kadi, 2022).

Research in this area has produced models that factor

in perfect and imperfect maintenance, as well as eco-

nomic and structural dependencies (Geng et al., 2015;

Gunn and Diallo, 2015). The literature on OM often

assumes component independence, with limited con-

sideration given to inter-component dependencies that

are critical in complex systems (Diallo et al., 2017).

The goal of this paper is to address this gap by incor-

porating economic, structural, and stochastic depen-

dencies into a comprehensive OM model.

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

180

2.3 Multi-Component System

Reliability Considering

Dependencies

Assessing the reliability of complex, interdependent

systems presents a formidable challenge. Stochastic

models, fault tree analysis (FTA), and Markov chains

are the primary methods employed in reliability esti-

mation (Wang et al., 2013; Zhang et al., 2021a; Fa-

zlollahtabar and Niaki, 2018; Bhangu et al., 2015).

(Kabir, 2017) highlighted the complexities involved

in FTA, necessitating the exploration of automated

methods for dependability synthesis. Markov mod-

els are another cornerstone technique used to evaluate

system states and transitions, exempliﬁed by (Ahmad

et al., 2022) and (Issa and Hassan, 2023). However,

the limitations of Markov modeling in repairable re-

dundant systems were pointed out by (Simpson and

Kelly, 2002).

An alternative approach involves copula functions

and Nataf transformations to model the joint distri-

bution of survival functions for components (Navarro

and Durante, 2017; Lin et al., 2021). The work of

(Xiao, 2014) and (Lin et al., 2020) has advanced this

methodology, enhancing the evaluation of system re-

liability in complex scenarios. In this paper, we adopt

these advanced techniques, applying Nataf transfor-

mation to map the stochastic dependencies of com-

ponent failures, informed by historical repair data,

thus advancing the reliability estimation for multi-

component systems within the CLSC context.

3 MODELING AND PROBLEM

FORMULATION

In this section, we deﬁne the problem setting and

describe the opportunistic maintenance optimization

model developed for managing spare parts within a

Closed-Loop Supply Chain (CLSC). Our approach

aims to balance operational costs and system reliabil-

ity, under the constraints of economic, stochastic, and

structural dependencies.

3.1 Problem Description

We consider a CLSC framework where failed medical

equipment units, consisting of various components,

are replaced in the ﬁeld with spare parts from an in-

ventory local to the client site. These failed units are

then sent to a repair center for thorough testing and

analysis. The objective is to identify and replace de-

fective components, and subsequently reintroduce the

refurbished spare parts into the supply chain for future

maintenance needs.

Within this system, maintenance decisions are

made opportunistically. Each time a unit fails and is

returned, the repair center is presented with a deci-

sion: to replace only the failed components or to also

proactively replace non-failed but aged components.

This decision aims to reduce the likelihood of future

failures and, in doing so, minimizes the long-term op-

erational costs while maintaining the reliability of the

spare parts within the CLSC.

This optimization problem is inherently dynamic;

each failure and subsequent repair decision resets

the life cycle of the affected components, creating

a new decision point. Unlike many existing models

that optimize the timing for preventive replacements,

this model introduces a single opportunity window to

make these decisions, aligning with real-world main-

tenance scenarios. The model accounts for three types

of dependencies that inﬂuence maintenance decision-

making:

• Economic Dependence: Where the collective

maintenance of a group of components incurs dif-

ferent costs compared to individual maintenance

actions, typically leading to cost savings when

multiple components are serviced simultaneously.

• Stochastic Dependence: Where the failure of one

component can have immediate, detrimental ef-

fects on the remaining components, potentially

accelerating their degradation or causing imme-

diate failure.

• Structural Dependence: Where components are

part of an interconnected assembly, necessitating

the disassembly of some parts to access others for

maintenance, as detailed in (Dinh et al., 2020).

This can impact both the duration of maintenance

activities and the condition of the components in-

volved.

The objective of this opportunistic maintenance op-

timization model is, then, to select a set of compo-

nents that need to be replaced preventively during

each maintenance opportunity. The model seeks to

minimize the maintenance cost, environmental im-

pact, and risk associated with spare parts repair, while

also minimizing the deviation from a predeﬁned reli-

ability threshold.

3.2 Mathematical Formulation

To formally deﬁne the optimization model, let us in-

troduce the following notations:

Integrating Reliability and Sustainability: A Multi-Objective Framework for Opportunistic Maintenance in Closed-Loop Supply Chain

181

Parameters:

• ζ = [1, 2, 3, ..n]: set of components in the spare

part,

• Cost

: price of component c,

• M

: the average lifetime of component c,

• RV

cost

: residual value of component c,

• LC: labor cost,

• Cost

: logistic cost for each repair (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 spare part as a function

of reliability of its components,

• f

sys

(t;a

, a

, .., a

): probability density function

of system’s lifetime,

• 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.

• S

price

: selling price of the spare part,

• R

min

: minimum required reliability.

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’ reliability

and ages. For example, for multi-independent units in

series, the reliability of the part can be expressed as

sys

(t) =

∏

c∈ζ

(t;a

Decision Variables: we deﬁne the binary decision

variable x

for each component c, with



1, if component c is replaced preventively,

0, otherwise.

Constraints: The model incorporates constraints

that ensure failed components are replaced correc-

tively 1 and the positive operational beneﬁt 2.

+ s

≤ 1, ∀c ∈ ζ. (1)

price

−C

maintenance

> 0. (2)

Objective Functions: The multi-objective opti-

mization problem aims to minimize the following

costs:

min {C

maintenance

, C

environement

, C

risk

, R

deviation

}

(3)

s.t. S

price

−C

maintenance

> 0,

+ s

≤ 1, ∀c ∈ ζ,

∈ {0, 1}, ∀c ∈ ζ.

Maintenance Cost (C

maintenance

): This cost in-

cludes expenses due to corrective and preventive re-

placements (5 as well as labor costs involved in the

maintenance process:

maintenance

= C

, (4)

where C

is the replacement cost calculated by sum-

ming the costs of components that are replaced either

correctively or preventively:

∑

c∈ζ

+ s

) ×Cost

. (5)

represents the labor costs, which are proportional

to the disassembly time required to replace the com-

ponents. We use an approach developed by (Dinh

et al., 2020) to calculate the total maintenance time

for a component group. Based on the structure con-

nection between components, the disassembly matrix

D is constructed. The elements of the matrix are bi-

nary coefﬁcients. The parameter D

i, j

= 1 if compo-

nent j must be disassembled to reach component 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 components 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. (3.2) gives the total

disassembly time, denoted by τ

group

, of the replaced

components:

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

replaced separately; the second term is the time sav-

ing due to intersections among the disassembly paths.

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182

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 in Eq. 3.2 equals 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)

Environmental Impact (C

environment

): The envi-

ronmental cost is associated with the waste of unused

remaining life of components that are replaced pre-

ventively:

environment

∑

c∈ζ

sys

(0;a

(1 − s

), .., a

(1 − s

))

(9)

+∞

t f

(t;a

)dt.

Risk Cost (C

risk

): The risk cost accounts for po-

tential failures within the planning horizon. 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 calcu-

lated. 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 reparation

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

))

(10)

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

f ailure

Cost

sys

(0;a

(1 − (x

+ s

)), .., a

(1 − (x

+ s

))

(11)

sys

(t; a

(1 − (x

+ s

)), .., a

(1 − (x

+ s

))

(1 + r)

dt.

Reliability Deviation (R

deviation

): This metric

quantiﬁes the deviation of the part’s reliability from

the required minimum. The reliability of the part

is deﬁned as the probability to survive the warranty

period T

warranty

given the components age after repair.

deviation

= 100 ×

max(R

min

−

sys

warranty

(1−(x

)),..)

sys

(0;a

(1−(x

)),..)

, 0)

min

(12)

4 SYSTEM RELIABILITY

MODELING CONSIDERING

STOCHASTIC DEPENDENCY

Computing the joint distribution of components’ life-

times is crucial for incorporating stochastic depen-

dency into optimization models. As reviewed in

Section 2, extant system reliability models that ac-

count for stochastic dependencies, such as Monte

Carlo simulations (Son et al., 2016; Issa and Has-

san, 2023; Ahmad et al., 2022), and copulas (Lin

et al., 2021; Navarro and Durante, 2017), are often

computationally intensive, rendering them impracti-

cal for high-dimensional problems. To surmount this

challenge, we introduce a dimensionality reduction

technique that computes the joint distribution of de-

pendent components’ lifetimes, subsequently utiliz-

ing Nataf’s transformation to accommodate the de-

pendencies among clusters of components.

4.1 Components Clustering

The construction of dependent clusters begins by cal-

culating covariance coefﬁcients between each pair of

components from historical repair data. A distance

matrix H = (h

i, j

) is derived from the correlation co-

efﬁcients R = (ρ

i, j

); (h

i, j

= 1 − |ρ

i, j

| ∀i, j). This

inverse relationship ensures that stronger dependen-

cies correspond to shorter distances. Components are

then clustered using the Agglomerative Hierarchical

Clustering algorithm, a common hierarchical cluster-

ing method that groups objects based on similarity

(Sasirekha and Baby, 2013).

The reliability distribution for each cluster is then

constructed, considering each cluster as an indepen-

dent ’super component’. The failure of any individual

component within a cluster implies the failure of the

entire cluster. The system reliability function under

the independence assumption is given by:

∗

sys

(t) =

∏

g∈ζ

(t;a

) (13)

We assess the conﬁgurations based on the likeli-

hood and the number of components per group, aim-

ing to select a threshold δ that minimizes group size

and maximizes the likelihood of the system’s time-

to-failure distribution. Analogous to the Akaike In-

formation Criterion (AIC), we formulate an index to

Integrating Reliability and Sustainability: A Multi-Objective Framework for Opportunistic Maintenance in Closed-Loop Supply Chain

183

evaluate the optimal solution. For a statistical model

with k parameters and a maximized likelihood func-

tion L, the AIC is AIC = 2k −2ln(L). The objective of

AIC is to minimize the number of parameters k while

maximizing the likelihood. In our case, we consider k

as the maximum number of components per group.

4.2 Nataf Transformation

Upon establishing the optimal grouping conﬁguration

as detailed in Section 4.1, the Nataf transformation

is employed to compute the joint distribution of each

component group. Pioneered by (Liu and Der Ki-

ureghian, 1986) and further innovated by (Lebrun

and Dutfoy, 2009), the transformation is a statisti-

cal method for dealing with correlated random vari-

ables by mapping them from their original distribu-

tion space to a standard normal space.

u = T

(X) = T

◦ T

(X) (14)

where :

: X → W = [F

), ..., F

)]

: W → Z = [Φ

−1

), ..., Φ

−1

)]

: Z → U = L

−1

(.) is the inverse CDF of the standard normal vec-

tor Z, F

is the CDF of the component x

, and L rep-

resents the lower triangular matrix obtained from the

Cholesky decomposition of R

= (ρ

i, j

) the correla-

tion matrix of the standard normal vector Z. The joint

probability of random vector X is then formulated as

follows:

P(X ≤t) = P(X

, X

, ..., X

≤ t)

= Φ

(Φ

−1

(t)), Φ

−1

(t)), ..., Φ

−1

(t)))

(15)

The correlation matrix of the random vector X, de-

noted by R

, is transformed into a standard normal

space using a linear search technique as described by

(Li et al., 2008; Xiao, 2014). The search is iteratively

reﬁned until the difference between b and a falls be-

low a predetermined error threshold ∆.

The joint probability density function for a group

of components in a serial system, represented by the

survival function R

(t) can be expressed as follows:

(t) = P(min

c∈g

) ≥ t) (16)

= P(

c∈g

≥ t)

= 1 − P(

[

c∈g

≤ t)

= 1 −

∑

c∈g

(t) +

∑

1≤c

≤c

≤n

P(X

, X

≤ t) + ...

+ (−1)

∑

1≤c

≤c

...≤c

≤n

P(X

, X

, ..., X

≤ t)

= 1 +

∑

1≤k≤n

(−1)

∑

1≤c

≤c

...≤c

≤n

P(X

, X

, ..., X

≤ t)

This approach simpliﬁes the computational process

and enables system reliability evaluation for high-

dimensional problems involving multiple dependent

components.

5 INDUSTRIAL CASE STUDY

We present in this section an application of the de-

veloped model based on a real industrial case from

our industrial partner GE Healthcare. GE Healthcare

(GEHC) is one of the global leaders in sales and ser-

vices of medical systems, notably those of medical

imaging with 4 million of millions systems installed

in more than 160 countries. Because of the criticality

of its products (medical devices) and the technologi-

cal characteristics of its components, GE Healthcare

offers a maintenance service to its customers. The

service’s main objective is to ensure its products’ re-

liability (reducing the failure rate occurrence) while

reducing unavailability simultaneously. As the spare

parts are expensive and also because of diposal the

failed parts directly will create circularity problems, a

CLSC is implemented in GEHC.

In this study, we implement the developed OM op-

timization model on the power supply of MRI ma-

chines, and investigate the strategy impact on the efﬁ-

ciency of the CLSC. The considered spare part com-

poses of 11 components connected in series. Figure 1

and Table 1 represent the physical structure between

components and the disassembling order and the nec-

essary operation times for each component, respec-

tively. For example, components 4, 5, and 7 must be

disassembled before component 10 can be replaced.

Table 2 represents the purchase price for new compo-

nents and their average useful life Mul in (U.T).

Table 1: Components’ dismantling time.

Component C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

Disassembling

time (U.T)

3 1 1.5 0.2 2 4.5 9 4.5 1 1 1

GEHC spare parts supply chain is so particular

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184

Figure 1: System’s structure.

Table 2: Costs parameters.

Component C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

Component

cost (U.C)

22 24 6 43 140 2 34 23 6 9 8

Mul (U.T) 71k 22k 54k 44k 1.5k 16k 68k 183k 37k 58k 24k

because its efﬁciency mainly depends on the qual-

ity of their repaired parts. The repair center provides

around one thousand of repaired power supply per

year to maintain Thousands of MRI systems. Be-

sides, depending on forecasted demand and stock lev-

els, one part can be reallocated multiple time to dif-

ferent warehouses around the world before being in-

stalled on a system which makes gathering parts’ life-

time challenging. Different information systems and

databases are used to store maintenance and logistic

data. We developed an algorithm to extract repair in-

formation and lifetime data for each replaceable unit

(LRU).

We have collected 13200 operating time and

maintenance records for 7514 spare parts. 3490 LRUs

were repaired multiple times. We consider the oper-

ating time of the functioning parts as censored. We

randomly selected 120 LRUs that was repaired multi-

ple times with different ages and multiple failed com-

ponents. We apply the optimization model on the se-

lected parts for a planning horizon of 730 Unit of time

(U.T). The remaining records are used to build System

reliability functions. Table 3 represents an example

of the data format. Column ID and Repair Number

are the part’s serial number and repair record. The

functioning parts are marked with value 1 in column

Censored. Column Time to failure represents the

observed time to event before each repair. The re-

maining columns represent the failed components. A

value equal to 1 for a component C

represents its fail-

ure.

In implementing the opportunistic maintenance

model delineated by equations (3), we estimate the

lifetime distributions from the data detailed in table

3 and take into account the stochastic dependencies

as detailed in section 4. Due to the complex nature

of the problem, an analytical solution remains elusive

for two primary reasons. Firstly, the failure cost eval-

uation in equation (11) requires calculating the con-

ditional reliability post-maintenance, which compli-

cates any attempt to linearize the objective function.

Moreover, the problem is classiﬁed as NP-hard be-

cause each component presents two potential scenar-

ios, leading to 2

possible conﬁgurations for a system

with n components. To navigate this computational

complexity and derive a near-optimal solution for the

opportunistic maintenance strategy, we employ the

Non-dominated Sorting Genetic Algorithm (NSGA).

This algorithm is particularly adept at handling multi-

objective optimization problems since it provides a

set of Pareto-optimal solutions representing the trade-

offs among the objectives. The decision-making pro-

cess will involve selecting the most appropriate solu-

tions based on the operational goals and constraints of

the CLSC.

6 RESULTS AND ANALYSIS

This section reports the results of our opportunistic

strategy on the test subset. First, we present the per-

formance of the clustering method described in sec-

tion 4 to model dependency in 6.1. Then, we present

the trade-off between different objectives and their

impact on decisions in section 6.2. Finally, we present

the opportunistic maintenance effect on parts’ relia-

bility in section 6.3.

6.1 Reliability Functions Under

Dependency

To build a system reliability model under dependency,

we select replacement records for the failed parts

and compute the correlation matrix between compo-

nents replacement using the Pearson method (Li et al.,

2012). Figure 2 shows the obtained correlation ma-

trix. Visually examining the results reveals that there

are different grouping possibilities. We can either

group C1, C2, with C3 and C7 with C10 or focus

on grouping only C1 with C3 and consider the re-

maining components independent. The grouping al-

gorithm presented in Section 4.1 is then used to de-

termine the best grouping strategy by maximizing the

likelihood and minimizing the number of components

per group. The results of the grouping algorithm are

given in Table 4, where threshold and groups rep-

resent the minimum correlation level selected and

the formed groups, respectively; N

super comp and

Max comp group are the number of formed groups

and the maximum number of components per group.

It can be seen from Table 4 that the best threshold for

Integrating Reliability and Sustainability: A Multi-Objective Framework for Opportunistic Maintenance in Closed-Loop Supply Chain

185

Table 3: Example of data format.

ID Repair Number Censored Time to failure C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

1 0 0 1260 0 0 0 1 1 0 0 0 0 0 0

1 1 0 1319 0 0 0 0 1 0 0 0 0 0 0

1 2 1 969 0 0 0 0 0 0 0 0 0 0 0

2 0 0 2159 0 0 0 0 1 0 0 0 0 0 0

2 1 1 1410 0 0 0 0 0 0 0 0 0 0 0

3 0 0 399 0 0 0 0 0 0 0 0 0 0 0

3 1 0 508 0 0 0 0 1 0 0 0 0 0 0

3 2 1 2628 0 0 0 0 0 0 0 0 0 0 0

grouping components is 0.6 because it maximizes the

loglikelihood and minimizes the number of compo-

nents per group. It can be veriﬁed that by grouping

like this, the resulting marginal distributions can sat-

isfy the positive deﬁnite constraint needed for apply-

ing Nataf transformation.

Figure 2: Correlation matrix.

The system reliability is then calculated based on

Eq. (16) considering the stochastic dependencies and

the forming clusters. Figure 3 shows the result and

compare it to the empirical estimations directly from

data. As we can see, the computed system lifetime

distribution from the proposed model ﬁts well the em-

pirical data.

6.2 Pareto Frontier Analysis and

Decision-Making Insights

The radar charts in ﬁgures 4 and 5 portray the Pareto

optimal solutions for two studied parts derived from

our multi-objective optimization framework. These

charts serve as a graphical elucidation of the trade-

offs between competing objectives within the context

of opportunistic maintenance scheduling.

Interpretive Analysis. Each axis on the radar

chart quantiﬁes an objective: maintenance costs

maintenance

), environmental impacts (C

environment

risk levels (C

risk

), net beneﬁts, the number of to-

tal replacements, and regulatory compliance penal-

ties (R

penalty

). The shape and reach of each solution’s

polygon on the chart indicate its performance across

these objectives. The farther a vertex extends from the

center, the higher the value in that speciﬁc objective,

thus facilitating a comparative analysis of the trade-

offs involved.

Equilibrium and Trade-Offs. The essence of the

Pareto frontier in a multi-criteria context is the bal-

ance between objectives. For instance, solutions that

extend towards the periphery for net beneﬁt illustrate

a preference for ﬁnancial optimization, potentially at

the expense of elevated risk. Meanwhile, solutions

with a more even distribution of vertices suggest a

more balanced approach, likely representing an equi-

librium amidst the conﬂicting objectives.

Managerial Implications. The model provides

profound insights into the strategic allocation of re-

sources for maintenance. It highlights the necessity

of a nuanced approach that transcends singular objec-

tive optimization:

• Solutions skewed towards net beneﬁt might res-

onate with proﬁt-maximizing agendas, albeit with

a vigilant eye on the rise in risk cost.

• Eco-centric solutions emphasize sustainability,

aligning with environmental compliance and so-

cial responsibility mandates.

• Risk-averse proﬁles cater to scenarios where the

cost of failure or downtime is prohibitive, under-

scoring the need for meticulous risk management.

• The minimization of R

penalty

reﬂects compliance-

centric strategies vital for adhering to regulatory

frameworks and avoiding ﬁscal penalties.

Conclusion. In conclusion, the Pareto frontiers un-

derscore the multifaceted nature of decision-making

in maintenance strategy optimization. Through a vi-

sual and quantitative articulation of trade-offs, the

model endows decision-makers to discern a balanced

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186

Table 4: Grouping components results.

Threshold Groups N super comp Max comp group Positive deﬁnite AIC Loglikelihood

0.6 (C1,C3) 10 2 1 10571.19 -10567.19

0.1 (C1,C2,C3,C7,C10) 7 5 0 10578.31 -10568.31

[0.4,0.5] (C1,C3); (C7,C10) 9 2 1 10583.59 -10579.59

[0.2,0.3] (C1,C2,C3);(C7,C10) 8 3 0 10591.78 -10585.78

[0.7,0.9] (C1);(C2);(C3);(C4);(C5);(C6);(C7);(C8);(C9);(C10);(C11) 11 1 1 10593.28 -10593.28

0 (C1,C2,C3,C4,C7,C10); (C5,C8,C11); (C6,C9) 3 6 0 26482.69 -26470.69

Figure 3: Modeling failure distribution under dependency.

Figure 4: Pareto frontier

for the ﬁrst case.

Figure 5: Pareto frontier

for the second case.

and holistic strategy that aligns with the broader spec-

trum of operational objectives.

6.3 Impact on the Reliability of the

Repaired Spare Parts

Figure 6 compares the reliability at the end of the hori-

zon plan T .Adopting a proactive maintenance strat-

egy has signiﬁcant implications for the reliability of

components after their horizon period. The model’s

performance is intrinsically linked to the manage-

ment’s risk tolerance. Preferences that lean towards

maximizing beneﬁts could adhere to the minimum

quality constraints, yet this approach carries an inher-

ent risk of failure, potentially leading to lower relia-

bility post-warranty.

Conversely, a strategy focused on minimizing risk

would enhance the reliability to its highest attainable

level, albeit at the cost of reduced beneﬁts. This con-

servative approach ensures a robust system that main-

tains performance beyond the horizon period, mitigat-

ing the likelihood of over-quality.

An average solution, representing a balance be-

tween the two extremes, offers a compromise that

aligns with intermediate managerial risk preferences.

This equilibrium point provides a performance be-

tween maximizing beneﬁts and minimizing risks.

This midpoint strategy can be particularly advan-

tageous for sustaining moderate reliability while

achieving reasonable beneﬁts, effectively balancing

the trade-offs between risk, cost, and reliability.

Figure 6: Impact of OM strategy on reliability at T

warranty

Decision-makers need to consider the long-term

implications of their chosen maintenance strategy on

the reliability of parts. As the warranty period con-

cludes, a maintenance plan overly focused on imme-

diate beneﬁts may result in increased costs due to fail-

ures and replacements. In contrast, prioritizing relia-

Integrating Reliability and Sustainability: A Multi-Objective Framework for Opportunistic Maintenance in Closed-Loop Supply Chain

187

bility can lead to sustained performance and reduced

long-term expenses, aligning with the overarching ob-

jectives of reliability-centered maintenance practices.

7 CONCLUSIONS AND

PERSPECTIVES

This research has presented a multi-objective opti-

mization framework for opportunistic maintenance

within Closed-Loop Supply Chains, focusing on the

medical systems domain. By integrating NSGA-

III, an advanced evolutionary algorithm, our model

handles the intricate balance of maintenance costs,

environmental impacts, risk, net beneﬁts, total re-

placements, and regulatory penalties. The empiri-

cal results, underpinned by real-world data from GE

Healthcare, have demonstrated the efﬁcacy of the pro-

posed model in navigating the complex trade-offs in-

herent in maintenance strategy optimization.

The Pareto frontiers elucidated through radar

charts visualize the trade-offs among competing ob-

jectives, empowering decision-makers to identify

strategies that align with their speciﬁc operational

goals and risk appetites. Whether the preference is

for cost minimization, risk aversion, or environmental

sustainability, the model offers the ﬂexibility to tune

the maintenance strategy accordingly.

We have also highlighted the chosen maintenance

strategy’s signiﬁcant impact on the parts’ reliability

after the warranty period. The preference towards

maximizing beneﬁts may lead to cost efﬁciencies in

the short term but can result in lower reliability post-

warranty. Conversely, a risk-averse approach that pri-

oritizes part reliability ensures system robustness but

may compromise on immediate ﬁnancial beneﬁts.

In summary, this study contributes a novel per-

spective to the literature on maintenance strategy opti-

mization within CLSCs and sets a precedent for future

research. Future work will aim to reﬁne the optimiza-

tion framework further and explore its applicability

across different industry sectors, thereby broadening

the impact of this research.

ACKNOWLEDGEMENT

We would like to thank Didier Rault and Stephane

Borrel from GE HealthCare for their helpful discus-

sions and results validation. The research of Zhiguo

Zeng is ﬁnancially supported by ANR under grant

number ANR-22-CE10-0004 and partially supported

by the chaire of Risk and Resilience of Complex Sys-

tems (Chaire EDF, Orange and SNCF). The partici-

pation of David Coit in this research is partially ﬁ-

nanced by the international visiting grant from Cen-

tralesup

elec, and the Bourses Jean d’Alembert from

Universit

e Paris-Saclay.

REFERENCES

Ab-Samat, H. and Kamaruddin, S. (2014). Opportunistic

maintenance (om) as a new advancement in mainte-

nance approaches: A review. Journal of Quality in

Maintenance Engineering, 20(2):98–121.

Agrawal, V. V., Atasu, A., and Van Wassenhove, L. N.

(2019). Om forum—new opportunities for operations

management research in sustainability. Manufactur-

ing & Service Operations Management, 21(1):1–12.

Ahmad, H. H., Almetwally, E. M., and Ramadan, D. A.

(2022). A comparative inference on reliability estima-

tion for a multi-component stress-strength model un-

der power lomax distribution with applications. AIMS

Math, 7:18050–18079.

Bhangu, N. S., Pahuja, G., and Singh, R. (2015). Appli-

cation of fault tree analysis for evaluating reliability

and risk assessment of a thermal power plant. Energy

Sources, Part A: Recovery, Utilization, and Environ-

mental Effects, 37(18):2004–2012.

Chen, H., Dong, Z., Li, G., and He, K. (2021). Re-

manufacturing process innovation in closed-loop sup-

ply chain under cost-sharing mechanism and different

power structures. Computers & Industrial Engineer-

ing, 162:107743.

Diallo, C., Venkatadri, U., Khatab, A., and Bhakthavatcha-

lam, S. (2017). State of the art review of quality, re-

liability and maintenance issues in closed-loop supply

chains with remanufacturing. International Journal of

Production Research, 55(5):1277–1296.

Dinh, D.-H., Do, P., and Iung, B. (2020). Maintenance op-

timisation for multi-component system with structural

dependence: Application to machine tool sub-system.

CIRP Annals, 69(1):417–420.

Esmaeili, N., Teimoury, E., and Pourmohammadi, F.

(2021). A scenario-based optimization model for

planning and redesigning the sale and after-sales ser-

vices closed-loop supply chain. RAIRO-Operations

Research, 55:S2859–S2877.

Fazlollahtabar, H. and Niaki, S. T. A. (2018). Fault tree

analysis for reliability evaluation of an advanced com-

plex manufacturing system. Journal of advanced

Manufacturing systems, 17(01):107–118.

Fleischmann, M., Bloemhof-Ruwaard, J. M., Dekker, R.,

Van der Laan, E., Van Nunen, J. A., and Van Wassen-

hove, L. N. (1997). Quantitative models for reverse

logistics: A review. European journal of operational

research, 103(1):1–17.

Geng, J., Azarian, M., and Pecht, M. (2015). Opportunis-

tic maintenance for multi-component systems con-

sidering structural dependence and economic depen-

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

188

dence. Journal of Systems Engineering and Electron-

ics, 26(3):493–501.

Gobbi, C. (2011). Designing the reverse supply chain: the

impact of the product residual value. International

Journal of Physical Distribution & Logistics Manage-

ment, 41:768–796.

Guide, V. D. R. and Wassenhove, L. N. (2003). Busi-

ness aspects of closed-loop supply chains, volume 12.

Carnegie Mellon University Press Pittsburgh, PA.

Gunn, E. A. and Diallo, C. (2015). Optimal opportunistic

indirect grouping of preventive replacements in mul-

ticomponent systems. Computers & Industrial Engi-

neering, 90:281–291.

Guo, Y., Shi, Q., and Guo, C. (2022). A fuzzy robust pro-

gramming model for sustainable closed-loop supply

chain network design with efﬁciency-oriented multi-

objective optimization. Processes, 10(10):1963.

Haque, S. A., Kabir, A. Z., and Sarker, R. A. (2003). Opti-

mization model for opportunistic replacement policy

using genetic algorithm with fuzzy logic controller.

In The 2003 Congress on Evolutionary Computation,

2003. CEC’03., volume 4, pages 2837–2843. IEEE.

Issa, L. A. A. A. and Hassan, Z. A. H. (2023). Application

of markov models to maintenance-required systems.

In AIP Conference Proceedings, volume 2591, page

050012. AIP Publishing LLC.

Jiang, R. and Ji, P. (2002). Age replacement policy: a multi-

attribute value model. Reliability Engineering & Sys-

tem Safety, 76(3):311–318.

Kabir, S. (2017). An overview of fault tree analysis and

its application in model based dependability analysis.

Expert Systems with Applications, 77:114–135.

Lebrun, R. and Dutfoy, A. (2009). An innovating analysis

of the nataf transformation from the copula viewpoint.

Probabilistic Engineering Mechanics, 24(3):312–320.

Li, D.-Q., Wu, S.-B., Zhou, C.-B., and Phoon, K. (2012).

Performance of translation approach for modeling

correlated non-normal variables. Structural safety,

39:52–61.

Li, H., L

u, Z., and Yuan, X. (2008). Nataf transformation

based point estimate method. Chinese Science Bul-

letin, 53(17):2586–2592.

Li, M., Jiang, X., and Negenborn, R. R. (2021). Op-

portunistic maintenance for offshore wind farms with

multiple-component age-based preventive dispatch.

Ocean Engineering, 231:109062.

Lin, S.-W., Matanhire, T. B., and Liu, Y.-T. (2021). Copula-

based bayesian reliability analysis of a product of a

probability and a frequency model for parallel systems

when components are dependent. Applied Sciences,

11(4):1697.

Lin, X., Jiang, Y., Peng, S., Chen, H., Tang, J., and Li,

W. (2020). An efﬁcient nataf transformation based

probabilistic power ﬂow for high-dimensional cor-

related uncertainty sources in operation. Interna-

tional Journal of Electrical Power & Energy Systems,

116:105543.

Liu, P.-L. and Der Kiureghian, A. (1986). Multivari-

ate distribution models with prescribed marginals and

covariances. Probabilistic engineering mechanics,

1(2):105–112.

Navarro, J. and Durante, F. (2017). Copula-based represen-

tations for the reliability of the residual lifetimes of

coherent systems with dependent components. Jour-

nal of Multivariate Analysis, 158:87–102.

Rebaiaia, M.-L. and Ait-Kadi, D. (2022). A remaining

useful life model for optimizing maintenance cost

and spare-parts replacement of production systems

in the context of sustainability. IFAC-PapersOnLine,

55(10):1562–1568.

Sasirekha, K. and Baby, P. (2013). Agglomerative hierar-

chical clustering algorithm-a. International Journal

of Scientiﬁc and Research Publications, 83(3):83.

Simpson, K. and Kelly, M. (2002). Reliability assessments

of repairable systems—is markov modelling correct?

In Safety and Reliability, volume 22, pages 19–39.

Taylor & Francis.

Son, K. S., Kim, D. H., Kim, C. H., and Kang, H. G. (2016).

Study on the systematic approach of markov modeling

for dependability analysis of complex fault-tolerant

features with voting logics. Reliability Engineering

& System Safety, 150:44–57.

Souza, G. C. (2013). Closed-loop supply chains: a crit-

ical review, and future research. Decision Sciences,

44(1):7–38.

Tahirov, N., Hasanov, P., and Jaber, M. Y. (2016). Opti-

mization of closed-loop supply chain of multi-items

with returned subassemblies. International Journal of

Production Economics, 174:1–10.

van Nunen, T. M. S. M. (1995). J van wassenhove l strategic

issues in product recovery management. California

Management Review, 372:114–135.

Wang, C., Xing, L., and Levitin, G. (2013). Reliability anal-

ysis of multi-trigger binary systems subject to compet-

ing failures. Reliability Engineering & System Safety,

111:9–17.

Wang, J., Miao, Y., Yi, Y., and Huang, D. (2021). An im-

perfect age-based and condition-based opportunistic

maintenance model for a two-unit series system. Com-

puters & Industrial Engineering, 160:107583.

Xiao, Q. (2014). Evaluating correlation coefﬁcient for nataf

transformation. Probabilistic Engineering Mechanics,

37:1–6.

Zhang, J., Kang, J., Sun, L., and Bai, X. (2021a). Risk

assessment of ﬂoating offshore wind turbines based

on fuzzy fault tree analysis. Ocean Engineering,

239:109859.

Zhang, X.-m., Li, Q.-w., Liu, Z., and Chang, C.-T.

(2021b). Optimal pricing and remanufacturing mode

in a closed-loop supply chain of weee under govern-

ment fund policy. Computers & Industrial Engineer-

ing, 151:106951.

Integrating Reliability and Sustainability: A Multi-Objective Framework for Opportunistic Maintenance in Closed-Loop Supply Chain

189