Single and Multiple Objective Optimization Models for Opportunistic

Preventive Maintenance

Carlos Henggeler Antunes

1

, Maria João Alves

2

, Joana Dias

2

, Teresa Gomes

1

, Benjamim Cardoso

3

and José Freitas

3

1

Department Electrical and Computer Engineering, University of Coimbra/INESC Coimbra,

Polo 2, 3030-290 Coimbra, Portugal

2

Faculty of Economics, University of Coimbra/INESC Coimbra, Av. Dr. Dias da Silva 165, 3004-512 Coimbra, Portugal

3

Critical Software, Parque Industrial de Taveiro, Lote 49, 3045-504 Coimbra, Portugal

Keywords: Opportunistic Maintenance, Binary Optimization, Multi-objective Models.

Abstract: This paper presents single and multiple objective deterministic optimization models for opportunistic

preventive maintenance of multi-component systems. The single objective model develops an aggregate

cost objective function encompassing costs of replacement, fixed costs associated with maintenance

interventions and costs of component dismounting whenever the replacement of a given component implies

disassembling others. Two multiple objective models are proposed, which enable to explore the trade-offs

between minimizing costs vs. the number of maintenance interventions and minimizing costs vs.

maximizing the remaining lifetime of components at the end of the planning period. Constraints refer to the

requirement of replacing each component before the end of its lifetime and consistency restrictions to allow

opportunistic maintenance and dismounting requirements.

1 INTRODUCTION

Maintenance generally encompasses the care and

servicing by specialized personnel for the purpose of

maintaining equipment and facilities in the required

operating conditions (Ben-Daya et al., 2009). The

primary goal of maintenance is to avoid or mitigate

the consequences of failure of equipment. The

optimization of maintenance operations and

schedule is of utmost importance in industry and

services, in particular in equipment-intensive

industries and utilities (e.g., aviation, energy,

telecommunications, water). Maintenance has a

direct impact on equipment reliability and

availability, and therefore on operational costs. Also,

adequate maintenance and facility management

policies and practices are central in sustaining safety

and eco-efficiency.

Corrective maintenance actions, i.e. those

performed after failure has occurred (run-to-failure),

may result in unwanted system disturbances such as

too frequent shutdowns with the consequent impacts

on costs and quality of service, and even on the

environment.

Preventive maintenance is aimed at preserving

the equipment operating conditions and preventing

their (otherwise costly) failure, involving partial or

complete overhauls to preserve and restore

equipment reliability. It provides for systematic

inspection, detection, and correction of emerging

failures before they happen or develop into major

faults. It may include tests, measurements,

adjustments and component replacement to prevent

faults from occurring. In general, preventive

maintenance is regularly or condition-based

performed on an equipment, or worn components,

often still working, to lessen the likelihood of

failing. Therefore, production loss, downtime, and

safety and environmental hazards are minimized.

Preventive maintenance is generally scheduled

based on a time or usage activation signal. An air-

conditioner is a typical example of an asset for

which a time-based preventive maintenance

schedule is performed: e.g., it is checked every year,

before the hot season. An example of an asset with a

usage-based preventive maintenance schedule is a

motor vehicle that should be scheduled for service

every 20,000 km. Applications that are generally

mentioned as suitable for preventive maintenance

Antunes, C., Alves, M., Dias, J., Gomes, T., Cardoso, B. and Freitas, J.

Single and Multiple Objective Optimization Models for Opportunistic Preventive Maintenance.

DOI: 10.5220/0005661601070114

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 107-114

ISBN: 978-989-758-171-7

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

107

include those that have a critical operational

function, failure modes that can be prevented with

regular maintenance, or a likelihood of failure that

increases with time or usage.

Since the maintenance schedule should be

planned, preventive maintenance is more complex to

coordinate than corrective maintenance. In this

scope, maintenance should be mainly seen as

investment in reliability and availability rather than a

cost-inducing activity. Therefore, optimization

approaches are required, aimed at encompassing the

essential features of real-world maintenance

problems in different settings, i.e. taking into

account the specificities of each industry and their

equipment usage, in order to generate optimal

recommendations to planners and decision makers.

In this setting, opportunistic maintenance models

are well suited to several real-world problems, thus

accommodating flexible strategies for planning

maintenance activities (Dekker et al., 1997).

Opportunistic models, i.e. preventive maintenance

activities at an opportunity, entail deciding whether

additional maintenance activities beyond the ones

that are strictly required should be performed at a

(possibly already planned) maintenance occasion.

I.e., if the system is already under maintenance

(either working or in a shutdown mode),

components may be replaced or maintained at no

additional fixed cost for intervention. Opportunistic

maintenance optimization using deterministic

models has been considered by Epstein and

Wilamowsky (1985), Dickman et al. (1990), Nilsson

et al. (2009) and Almgren et al. (2012), among

others. Stochastic opportunistic replacement models

have also been studied by several authors including

the recent work by Patriksson et al. (2015).

In this paper single and multiple objective

deterministic mathematical models are developed to

provide decision support in the scheduling of

opportunistic maintenance activities. Decisions to be

made involve component replacement and

component dismounting whenever the replacement

of a given component implies disassembling others.

The single objective function aggregates these

different costs to determine the optimal solution.

The single objective model has been developed from

the basic opportunistic replacement model by

Almgren et al. (2012) by including component

dismounting actions.

Two models with multiple objective functions

are then proposed, which enable to explore the trade-

offs between minimizing replacement and

dismounting costs vs. minimizing the number of

maintenance interventions and minimizing total

costs vs. maximizing the remaining lifetime of

components at the end of the planning period. For

multi-objective models the nondominated (Pareto

optimal) set is computed. A feasible solution is

nondominated if no other feasible solution exists that

simultaneously improves all objective function

values, i.e. improving an objective function implies

worsening the value of at least another objective

function value. In the present work the whole

nondominated set has been obtained using a

procedure based on reference points co-developed

by one of the authors (Alves and Clímaco, 2000).

In section 2 a single objective model for

optimizing opportunistic maintenance is presented,

in which an overall cost objective function is

considered. Multi-objective models for decision

support in opportunistic maintenance are presented

in section 3. Conclusions are drawn and further

research is outlined in section 4.

2 A SINGLE OBJECTIVE MODEL

FOR OPTIMIZING

OPPORTUNISTIC

MAINTENANCE

In this section a single objective mathematical model

devoted to optimize the maintenance scheduling of a

multi-component system is presented. The

components must be replaced before they reach the

end of their lifetime; this is estimated so that the

probability of a component failure within its lifetime

is low enough for its intended use and is usually

provided by the equipment manufacturer. Hence, a

deterministic model is considered under this

assumption.

Considering a set of components and a finite

planning horizon discretized in time intervals, the

model aims at determining the dismounting and

replacement schedule of the components during the

planning horizon in order to minimize the total cost.

A component that is replaced must be firstly

dismounted. In addition, the replacement of a given

component may imply dismounting others in which

the component is embedded, regardless of whether

those components require or not maintenance at that

time interval.

Opportunistic maintenance is mainly justified

when there is a significant fixed cost associated with

a maintenance intervention, which is independent of

the components that are replaced. The proposed

model considers an overall cost objective function

including terms related to fixed (opportunity) costs

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

108

for interventions and costs for component

replacement and dismounting.

Constraints refer to:

- the requirement of replacing each component

before the end of its lifetime, including considering

that at the beginning of the planning period some

components may be already worn out (i.e. having

some time of use);

- enforcing the consideration of the fixed cost for

intervention if at least one component is replaced at

a given time interval (to induce maintenance

opportunities at no additional fixed cost);

- requirement of dismounting a component if it

contains another component that is replaced.

2.1 Single Objective Model

The model inputs are:

- A set of N components, which are the target of

the replacement/maintenance actions.

- A set of T time intervals, which result from the

discretization of the finite planning horizon.

- A maximum replacement interval L

i

for each

component i ∈{1,…,N} corresponding to its

estimated lifetime (this maximum replacement

interval can also derive from a policy decision, a

safety regulation associated with the component’s

technical life, or a contractual requirement).

L

0i

is the maximum replacement interval for each

component i for the first time in the planning period

T, thus taking into account its time of use before

t =0.

- The replacement cost c

it

of component i

∈{1,…,N} at time t ∈{1,…,T}.

- The fixed cost associated with a maintenance

intervention (opportunity cost) d

t

≥ 0 at time t

=1,…,T, which is independent of the number of

components replaced.

- The replacement of a component implies that it

should be firstly dismounted. The dismounting cost

of component i at time t is a

it

.

- If component i is embedded into other

component(s) then it may happen that in order to

dismount component i it is necessary to dismount

other component(s) as well. Let M(i) be the set of

components j that should be dismounted when

component i is dismounted.

Decision variables:

x

it

= 1, if component i ∈{1,…,N} is replaced at

time t ∈{1,…,T}; 0, otherwise.

y

it

= 1, if component i ∈{1,…,N} is dismounted

at time t ∈{1,…,T}; 0, otherwise.

w

t

= 1, if at least one replacement operation

occurs at time t ∈{1,…,T}; 0, otherwise.

The single objective model is:

Model S1

∑∑

==

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++

T

t

N

i

ttitititit

wdyaxc

11

)(min

(1)

subject to:

∑

=

=≥

i

L

t

it

Nix

0

1

,...,1 ,1

(2)

∑

+

+=

=−=≥

i

Lk

kt

iit

NiLTkx

1

,...,1 ;,...,1 ,1

(3)

NiTtwyx

titit

,...,1 ;,...,1 , ==≤≤

(4)

M(i)jTtyy

jtit

∈=≤ ;,...,1 ,

,∀i:M(i)≠∅

(5)

NiTtwyx

titit

,...,1 ;,...,1 }1,0{,, ==∈

(6)

The objective function (1) minimizes the total

cost considering the component replacement,

component dismounting and fixed costs for

interventions. Constraints (2) ensure that each

component is replaced within its maximum

replacement interval for the first time and (3) ensure

the replacement of each component before the end

of its lifetime for the rest of the planning period.

Constraints (4) ensure that a component is

dismounted before it is replaced and an intervention

operation occurs at that time interval. Constraints (5)

impose that each component j in which i is

embedded is also dismounted if i is dismounted.

The model (1)-(6) can be simplified to have

fewer variables and constraints, and thus minimize

the computational effort.

Let

∪

N

i

iMM

1

)(

=

=

.

The y

jt

variables will be defined only for j ∈ M.

Therefore, consider the following definitions:

x

it

= 1, if component i ∈{1,…,N} is dismounted

and replaced at time t ∈{1,…,T}; 0, otherwise.

y

jt

= 1, if component j ∈ M is dismounted at time

t ∈{1,…,T}; 0, otherwise.

w

t

keeps the same definition as above.

The simplified model is:

Model S2

∑

∑∑ ∑∑

=

== ∈∉

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++

T

t

tt

T

t

N

iMj

jtjt

Mi

itititit

wd

yaxaxc

1

11

min

(7)

subject to:

∑

=

=≥

i

L

t

it

Nix

0

1

,...,1 ,1

(8)

Single and Multiple Objective Optimization Models for Opportunistic Preventive Maintenance

109

∑

+

+=

=−=≥

i

Lk

kt

iit

NiLTkx

1

,...,1 ;,...,1 ,1

(9)

NiTtwx

tit

,...,1 ;,...,1 , ==≤

(10)

MjTtyx

jtjt

∈=≤ ;,...,1

(11)

M

(i)jTtyx

jtit

∈=≤ ;,...,1 ,

,∀i:M(i)≠∅

(12)

, , {0,1} 1,..., ; 1,..., ;

it t jt

x

wy t T i Nj M∈= = ∈

(13)

The model (7)-(13) has the same number of

variables and constraints as the model (1)-(6) only if

|M|=N. Otherwise it has fewer variables and

constraints. Constraints (10) and (11) correspond to

constraints (4) in Model S1 and constraints (12)

correspond to (5). Remind that x

it

= 1 means that

component i is dismounted and replaced at time t, so

variables y

jt

can be directly related to x

it

for which y

it

have not been defined (i.e., for i ∉ M). The fact that

y variables are not defined for all components leads

to a different formulation of the cost function (7),

associating the dismounting cost with y

jt

for

components j∈M and with x

it

for i ∉ M.

As only superfluous variables (and related

constraints) are eliminated from Model S1 to Model

S2, the two models are equivalent.

2.2 Illustrative Example

Model S2, (7)-(13), has been instantiated with the

following data for illustrative purposes: N=5

components and T=50 time intervals.

Table 1 displays the lifetime (L

0i

and L

i

), costs

for each component (c

it

and a

it

) and dismounting

requirements (M(i)).

Experiments have been carried out with fixed

costs for interventions d

t

= 10, 100, 1000, for all t.

Table 1: Lifetime, costs for each component (costs are the

same for all time intervals) and dismounting requirements.

Component 1 2 3 4 5

L

0i

2 5 11 4 15

L

i

7 10 16 9 20

c

i

t

80 185 160 125 150

a

i

t

20 45 40 30 35

M(i)

∅ ∅

1 2, 5

∅

These instances have 450 binary variables and

743 constraints. The equivalent Model S1 would

lead to instances with 550 binary variables and 843

constraints.

Tables 2-4 present optimal solutions for d

t

= 10,

100, 1000, respectively. “x” denotes component

replacement and “o” denotes dismounting without

replacement. These solutions were obtained using

our software for MultiObjective Mixed-Integer

Linear Programming (MOMILP) problems (see

section 3), which uses the non-commercial lpsolve55

software (http://lpsolve.sourceforge.net/5.5/) to

solve the integer single-objective problems. The

computation times for obtaining the optimal

solutions to the problems with d

t

= 10, 100 and

1000, in a computer with Intel Core i7-2600K

CPU@3.4GHz and 8 GB RAM, were 0.05, 2.65 and

0.08 seconds, respectively.

Table 2: Optimal solution for d

t

= 10.

Component →

Interval↓

1 2 3 4 5

2 x

4 x x o

9 x x

13 x x x

16 x o x o

23 x x x

24 o x o

30 x

33 x x x

37 x x

42 x x o

44 x

The total cost of the optimal solution for d

t

= 10

(Table 2) is 4100, 3980 for dismounting and

replacing plus 120 for intervention fixed cost, with

12 maintenance interventions being made.

Table 3: Optimal solution for d

t

= 100.

Component →

Interval↓

1 2 3 4 5

2 x

4 x x x o

10 x x

13 x x x

17 x

22 x x o

24 x x

31 x x x x

38 x o x x o

41 x

44 x o x o

The total cost of the optimal solution for d

t

= 100

(Table 3) is 5180, 4080 for dismounting and

replacing plus 1100 for intervention fixed cost, with

11 maintenance interventions.

The total cost of the optimal solution for d

t

=

1000 (Table 4) is 11690, 4690 for dismounting and

replacing plus 7000 for intervention fixed cost, with

7 maintenance interventions.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

110

Table 4: Optimal solution for d

t

= 1000.

Component →

Interval↓

1 2 3 4 5

2 x x x o

9 x x x x x

16 x x x o

23 x x x x x

30 x x x o

37 x x x x x

44 x x x o

The total cost of the optimal solution for d

t

=

1000 (Table 4) is 11690, 4690 for dismounting and

replacing plus 7000 for intervention fixed cost, with

7 maintenance interventions.

We can observe from these solutions that there is

frequent disassembling of component 5 due to the

frequent replacement of component 4, which has a

short lifetime. Although the replacement of

component 3 requires disassembling the component

1, it is not necessary to disassemble component 1

without its replacement in any solution because this

component has a short lifetime and low replacement

cost. Finally, it should be mentioned that solutions

for d

t

=10 and d

t

=100 (Tables 2-3) have several

optimal alternative solutions, i.e. different

combinations of component replacement and

disassembling in the planning period leading to the

same optimal objective function value.

This model requires the specification of fixed

costs for interventions d

t

, which in some cases may

be difficult to estimate. In order to better explore

compromises between cost (of replacement and

dismounting) and number of maintenance

interventions, a bi-objective model is proposed

below.

3 MULTIPLE OBJECTIVE

MODELS FOR DECISION AID

IN OPPORTUNISTIC

MAINTENANCE

In this section bi-objective models are proposed,

which enable to explore the trade-offs between

minimizing replacement and dismounting costs vs.

the number of maintenance interventions (Model

M1) and minimizing total costs vs. maximizing the

remaining lifetime of components at the end of the

planning period (Model M2). The results of the

illustrative examples were obtained using the

interactive MOMILP software co-developed by one

of the authors in Delphi for Windows (Alves and

Clímaco, 2004), which includes a reference point-

based procedure that is able to compute the whole

nondominated front for bi-objective problems.

3.1 Minimizing Costs Vs. Number of

Maintenance Interventions

The first objective function (14) of Model M1

minimizes the replacement and dismounting cost

while the second objective function (15) aims at

minimizing the number of maintenance

interventions.

Model M1

∑∑ ∑∑

== ∈∉

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++=

T

t

N

iMj

jtjt

Mi

itititit

yaxaxcF

11

1

min

(14)

∑

=

=

T

t

t

wF

1

2

min

(15)

subject to:

(8) – (13)

Example:

Model M1 has been instantiated with the same

data as the single-objective model. The resulting

problem has 7 nondominated solutions, which are

depicted in Figure 1.

Figure 1: Nondominated front: replacement and

dismounting cost vs. number of interventions (Model M1).

The nondominated solution that minimizes cost,

(F

1

, F

2

) = (3980, 12), is an alternative optimal

solution to the single objective model with the

smaller value of d

t

, i.e. d

t

=10. The nondominated

solution that minimizes the number of maintenance

interventions, (F

1

, F

2

) = (4690, 7), is the optimal

solution to the single objective model with d

t

=1000.

The solution where (F

1

, F

2

) = (4080, 11) is an

alternative optimal solution for the single objective

model with d

t

=100. In addition to these solutions,

there are other intermediate nondominated solutions

with 8, 9 and 10 interventions. Note that several

alternative configurations (decision variable values)

Single and Multiple Objective Optimization Models for Opportunistic Preventive Maintenance

111

can be found for the same nondominated objective

point. This happens, in particular, for the solutions

that consider a large number of maintenance

interventions. However, the nondominated points

(objective function values) are only those shown in

Figure 1.

3.2 Minimizing Total Cost Vs.

Maximizing the Remaining

Lifetime of Components at the End

of the Planning Period

In addition to the minimization of maintenance

costs, the maximization of the value of the assets at

the end of the planning period may also be an

objective the decision maker wants to accomplish,

namely if the planning period should be extended

due to any circumstance. The real value of a

component depends on its remaining lifetime. The

next model (Model M2) considers again the overall

cost (replacement and dismounting plus the

intervention fixed cost) for the first objective

function. Thus, its formalization (16) is the same as

the objective function (7) of the single objective

model. The second objective function aims at

maximizing the remaining lifetime of the

components, as a proxy for maximizing the value of

the assets at the end of the planning period. As the

decision maker may want to assign different levels

of importance to each component, a weighted sum of

the remaining lifetime of the components is

considered. This objective function is formalized in

(17), where α

i

denotes the weight assigned to each

component i.

If the component i is replaced at the last time

interval of the planning period, t=T, then its

remaining lifetime is L

i

; if the replacement is at

t =T-1, then its remaining lifetime is L

i

– 1, and so

on; thus the remaining lifetime of the component at

the end of the planning period can be given by

∑

+−=

T

LTt

itit

i

xv

1

with

kLv

iit

−=

for

kTt −=

, provided

the model ensures that the component is replaced

only once from the time

i

LT −

to T. Accordingly,

the replacement constraints of each component for

the last period, i.e. constraints (9) for

i

LTk −=

,

are changed to be of type “=” instead of “≥”. These

are constraints (19) in Model M2; the other

constraints of (9) are replaced by (18) in this model.

Model M2

∑

∑∑ ∑∑

=

== ∈∉

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++=

T

t

tt

T

t

N

iMj

jtjt

Mi

itititit

wd

yaxaxcF

1

11

1

min

(16)

∑∑

=+−=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

α=

N

i

T

LTt

ititi

i

xvF

11

2

max

(17)

subject to:

(8), (10) – (13)

∑

+

+=

=−−=≥

i

Lk

kt

iit

NiLTkx

1

,...,1 ;1,...,1 ,1

(18)

∑

+−=

==

T

LTt

it

i

Nix

1

,...,1 ,1

(19)

Examples:

Model M2 has been instantiated with the same

data as the previous models using d

t

=100.

Two experiments were performed. The first one

considered α

i

=1 for all i. In the second experiment

higher weight was given to components with smaller

lifetime: α

i

=1/L

i

. The weights were then normalized

so that

N

N

i

i

=α

∑

=1

. The normalization enables a

better comparison between the two experiments as

the weights have equal sum.

In the first experiment 23 nondominated

solutions were obtained, which are depicted in figure

2. The overall cost ranges from 5180 to 6230. The

solution with minimum cost (solution A in Figure 2,

shown in Table 5) presents a sum of remaining

lifetime at the end of the planning period of 14.

Table 5 also shows the remaining lifetime of each

component. This solution is an alternative optimal

solution to the single objective Model S2 with

d

t

=100. However, the optimal solution of Model S2

presented in Table 3 has a sum of remaining lifetime

of only 9. Solution A is also an alternative to the

nondominated solution that minimizes cost in the bi-

objective Model M1 (cost vs. number of

maintenance interventions). Likewise, the number of

maintenance interventions is equal to 11. However,

the sum of remaining lifetime is 12 in the solution

obtained for Model M1, which is worse than the

corresponding value in solution A for Model M2.

Hence, the nondominated solution that minimizes

cost obtained for Model M2 may be more interesting

than the solutions that minimize cost for the single

objective Model S2 and the multiobjective Model

M1: all these solutions have the same cost and

number of maintenance interventions but solution A

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

112

presents a larger overall remaining lifetime at the

end of the planning period.

The nondominated solution that maximizes the

sum of remaining lifetime (solution B in Figure 2,

shown in Table 6) has a value of 62 for this

objective and a total cost of 6230. This solution

proposes the replacement of all components at the

last time interval of the planning period, i.e. t =

T =50, so it ensures the maximum remaining lifetime

for all components.

Figure 2: Nondominated front: overall cost vs. sum of

remaining lifetime at the end of the planning period

(Model M2 – experiment 1).

Table 5: Solution A that minimizes cost in Model M2.

Component →

Interval↓

1 2 3 4 5

1 x o x o

5 x

8 x o x x o

15 x x x x

22 x x

24 x x o

29 x

33 x x x

36 x x

42 x x x o

49 x

remaining lifetime 6 2 2 1 3

In the second experiment, considering a

weighted-sum for the remaining lifetime objective

function, 31 nondominated solutions were obtained,

which are depicted in Figure 3. The extreme

solutions (A and B), which optimize individually

each objective function, are similar in both

experiments. The problem solved in the second

experiment has more nondominated solutions than

the one solved in the first experiment (with equal

weights) and some solutions are common, but not all

solutions of the first problem belong to the

nondominated set of the second problem.

Table 6: Solution B that maximizes the sum of remaining

life in Model M2.

Component →

Interval↓

1 2 3 4 5

2 x x x o

9 x x

11 x x x

13 x

20 x x x x o

27 x o x o

30 x x

34 x o x x o

40 x x

41 o x o

43 x

50 x x x x x

remaining lifetime 7 10 16 9 20

An intermediate solution obtained in these

experiments with Model M2 can also be analyzed,

e.g. solution C (Figures 2 and 3), common to both

experiments. The solution configuration is presented

in Table 7. It requires 12 maintenance interventions.

The total cost is 5605 and the sum of remaining

lifetime is 44 (the weighted sum of remaining

lifetime is 39.89 – value of F

2

in the second

experiment).

Figure 3: Nondominated front: overall cost vs. weighted

sum of remaining life at the end of the planning period

(Model M2 – experiment 2).

We can observe in Table 7 that four of the five

components (1, 2, 4 and 5) are replaced near the end

of the planning period, i.e., at t=49, so their

remaining lifetime is near the respective maximum

(L

i

−1). However, the last replacement of component

3 is at t=36, so its remaining lifetime at the end of

the planning period is only 2. Therefore, although

intermediate solutions exist displaying a good trade-

off between cost and (weighted) sum of remaining

lifetime at the end of the planning period, the

individual remaining lifetimes may not be balanced

among the various components. Solution D in Figure

Single and Multiple Objective Optimization Models for Opportunistic Preventive Maintenance

113

3 also needs 12 maintenance interventions. The last

replacement of each component occurs at t=48,

leading to a remaining lifetime of 5, 8, 14, 7 and 18,

respectively for components 1 to 5. The minimum

remaining lifetime is 5, thus the remaining lifetimes

are more balanced than in solution C. However,

solution D presents a higher cost than C (5770 vs.

5605).

Table 7: Solution C to Model M2.

Component →

Interval↓

1 2 3 4 5

1 x

4 x x o

8 x x

13 x x x

15 x

22 x x x x o

29 x x

31 x x o

36 x x

40 x x o

43 x

49 x x x x

remaining lifetime 6 9 2 8 19

4 CONCLUSIONS

Single and multiple objective deterministic

optimization models have been presented to support

opportunistic preventive maintenance decisions

regarding a set of components, which are the target

of the replacement/maintenance actions, over a finite

planning horizon. The single objective model

considers the minimization of an overall cost

objective function including fixed costs for

interventions and costs for component replacement

and dismounting whenever the replacement of a

given component implies disassembling others. The

multi-objective models enable to explore the trade-

offs between minimizing replacement and

dismounting costs vs. the number of maintenance

interventions and minimizing total costs vs.

maximizing the remaining lifetime of components at

the end of the planning period. Illustrative examples

have been presented using software developed by

some of the authors for general multiobjective

mixed-integer programming problems to exploit the

practical insights offered by the models.

Further research will involve developing

adequate sensitivity analysis techniques to take into

account the uncertainty associated with the model

coefficients to obtain robust solutions, i.e.

recommendations that are relatively immune to

changes of coefficients within plausible ranges.

Moreover, the models will be exploited in real-world

settings taking into account the particularities raised

by specific situations as well as scalability issues to

tackle large-scale problems in systems with

hundreds of components.

ACKNOWLEDGEMENTS

The authors acknowledge the support of iAsset

Project, ref. 38635/2014, financed by POFC, and

FCT project grants UID/MULTI/00308/2013 and

MITP-TB/CS/0026/2013.

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