A GENERALIZED EXTREMAL OPTIMIZATION-INSPIRED

ALGORITHM FOR PREDICTIVE MAINTENANCE

SCHEDULING PROBLEMS

Pasquale Arpaia

Dipartimento di Ingegneria, Università del Sannio, CERN European Laboratory for Nuclear Research

Department of Technologies, Group of Magnets, Superconductors and Cryostats

M26220, (30-03-030) CH-1211 Geneva 23, Switzerland

Domenico Maisto

Institute for High-Performance Computing and Networking (ICAR), Italian National Research Council (CNR)

Via P. Castellino 111, 80131 Naples, Italy

Carlo Manna

Dipartimento di Ingegneria Elettrica, Università degli Studi di Napoli Federico II

Via Claudio 21, 80125 Napoli, Italy

Keywords: Artificial Intelligence, Optimization Methods, Maintenance, Scheduling.

Abstract: A bit-encoded heuristic evolutionary optimization algorithm inspired by the Generalized Extremal

Optimization method is presented. The proposed evolutionary approach aims at optimizing a predictive

maintenance scheduling problem characterized by an analytically intractable objective function. A

preliminary comparison with a standard genetic algorithm on a set of high-dimension cases of the

considered maintenance problem shows better performance for the proposed approach.

1 INTRODUCTION

Evolutionary algorithms are excellent heuristic

methods, inspired by biological evolution, to solve

complex optimization problems with analytically

intractable objective functions. Although

evolutionary-based methods approximate the

optimal solution without guaranteeing its optimality,

the underlying principles of natural evolution ensure

promising results (De Sousa and Ramos, 2002). This

turns out to be useful especially in real-time

complex optimization.

The most popular and used methods are mainly:

Genetic Algorithms (GA) (Goldberg, 1989),

Simulated Annealing (SA) (Kirkpatrick et al, 1983),

and algorithms based on Swarm Intelligence, such as

Ant Colony Optimization (ACO) (Dorigo et al,

1996), and Particle Swarm Optimization (PSO)

(Kennedy and Eberhart, 1995), (although the last

two are biological inspired heuristics, not considered

tightly evolutionary by the survey).

However, the aforementioned algorithms in their

practical implementation for optimization problems

have a problematic feature: the optimal solution is

searched through a stochastic process sensitive to a

suitable setting of adjustable parameters. A proper

setting affects the performance of the algorithms

significantly, and in many practical cases this

becomes a costly task in itself. Moreover, most of

them are population-based, thus their run is time-

consuming compared to other algorithms.

By exploiting the Self-Organized Criticality state

theory (SOC) (Bak, Tang and Wiesenfeld, 1987) in

ecosystems, Boettcher and Percus proposed a novel

evolutionary optimization method called Extremal

Optimization (EO) (Boettcher and Percus, 2001),

successfully applied to complex combinatorial

optimization problems. EO method relies on the

Arpaia P., Maisto D. and Manna C..

A GENERALIZED EXTREMAL OPTIMIZATION-INSPIRED ALGORITHM FOR PREDICTIVE MAINTENANCE SCHEDULING PROBLEMS.

DOI: 10.5220/0003080200700076

In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 70-76

ISBN: 978-989-8425-31-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

Bak-Sneppen model (Bak and Sneppen, 1993), a

simplified model of natural co-evolution in

ecosystems: a number of species in a system evolves

to reach the best adaptation; the worst adapted

species are forced to evolve more quickly to avoid

extinction. This mechanism determines an overall

adaptation for the ecosystem as a whole.

Beyond these encouraging results, the

evolutionary approach proposed in (Boettcher and

Percus, 2001) adds two peculiar features: only one

setting parameter is needed and a single candidate is

processed at each iteration. These two aspects are “a

priori” advantages with respect to the traditional

evolutionary approach (as GA, SA, PSO and so on).

These noteworthy characteristics have

encouraged the employment of EO algorithm to

tackle different physics issues or engineering

applications, particularly hard to face.

Predictive maintenance scheduling belongs to

this class of problems; it could be described in this

way: an optimal action sequence for maintaining a

system in order to avoid potential breakdowns is to

be found. The terms predictive indicates that some

problem parameters cannot be constant during the

process, but are continuously updated in real time.

Thus, the planned schedule (the optimal solution)

must to be re-organized for every modification of

the examined system state and the constraints of the

task. Moreover, such as many maintenance

scheduling problems, the corresponding

optimization problem is characterized by an

analytically intractable objective function to be

minimized. Hence, it needs for a heuristic approach

to search the optimal solution.

Among the above variations of EO, the

Generalized Extremal Optimization (GEO)

algorithm (De Sousa, Ramos, 2002) was built to be

applied on a wide class of complex problems. Its

particularity lies in working on strings composed by

bits with “fitness” proportional to the contribution to

the quality of the whole solution generated by their

mutation.

Following this simple idea, in this paper a GEO

application is proposed for the problem of the

predictive maintenance. After an outline of the

proposed method, preliminary experimental results

on a set of analytically intractable scheduling

problems are shown in order to highlight better

performance than a standard GA.

2 THE PROPOSED METHOD

In the present section, first, a formulation of

predictive maintenance scheduling problem is

detailed and, then, the proposed heuristic algorithm

is presented.

2.1 Statement of the Predictive

Maintenance Scheduling Problem

2.1.1 Experimental Motivations

The maintenance scheduling formulation proposed

in the following is to be faced under the framework

of the industrial research project MONDIEVOB

(Buildings Remote Monitoring and Evolutionary

Diagnostics), granted by POR 3.17 ICT Regione

Campania (Italy).

The long-term goal of MONDIEVOB is a

predictive maintenance tool for processing

experimental information acquired from building to

be maintained in order to assess reliability and

predict possible future failures (Figure 1), by means

of algorithms able to predict future status of a

machine or a process (Stapelberg, 2009). This

predictive information allows proactive

responsiveness in maintenance decision-making.

Figure 1: “Model of failure” module predicts probability

of failure of the considered system, from past and present

data. This predictive information updates the objective

function of the maintenance scheduler, in real time.

Essentially, the on-line available information

about the status of the monitored systems allows

maintenance operations to be anticipated/delayed

according to the actual conditions.

In order to accomplish this task, a formulation

evaluating different maintenance scenarios by

considering the associated cost effects of the

resulting maintenance operations and taking into

account the current and predicted machine

degradation levels has been set up. The cost of

A GENERALIZED EXTREMAL OPTIMIZATION-INSPIRED ALGORITHM FOR PREDICTIVE MAINTENANCE

SCHEDULING PROBLEMS

maintenance actions, availability and maintenance

resource constraints are taken into account.

2.1.2 Evaluation of Maintenance Schedule

Effects

The purpose of the method presented in this section

is defining a cost function in order to evaluate the

effects of any given maintenance operation.

The cost function used here takes into account

both: the cost associated to the maintenance action

(as, for example, the replacement of a given

component), and the cost associated to the system

operating in the normal state (as monitoring,

inspection and so on).

Let n be the available resources to maintenance

operation, and m

(for i=1,..., M) the i-th system

component that must be maintained (for a total of M

components). The function C, representing the total

cost of planned maintenance, can be expressed as:

C = a

+ p

(t)*B

()

i∈G

∑

+ b

(t)

()

j∈H

∑

⎛

⎝

⎜

⎞

⎠

⎟

t =1

∑

(1)

in which the following notation is used:

T finite time horizon of planned maintenance

t for t = 1, ..., T, the t-th instant of the time

horizon T

the operating cost of the i-th component

the replacement cost for the j-th component

time dependent maintenance cost of the j-th

component (Dekker et al., 1997)

(t) probability of failure of the i-th component at

the time t

cost of breakdown of the i-th component

the set of every component not maintained at

the time t

the set of every maintained component at the

time t

Moreover, any given planned maintenance evaluated

by means of (1) is subject to the following

constraints:

(i) Each m

can be served (maintained) by only one

of the n available resources at any time t;

(ii) Each m

has to be served at least one time

instant t during the total time horizon T;

Finally, it should be noted that the probability of

failure p

at the time t could be derived from various

deterioration models (Djurdjanovic et al., 2003;

Engel et al., 2000; Yu et al, 2005), depending on the

type of monitored component, and from the nature

of information or signals acquired.

2.1.3 Bit encoded Solution

In the present work, each maintenance schedule S

(called sequence, in the following) evaluated by

means of (1) is expressed through a binary string

representation as:

{

}

TMTTM

ssssssS ,,,;;,,,

2111211

…………

(2)

where s

is the value of the corresponding bit. For

example, s

=1, means that the 3-th component is

maintained at the time instant t =1.

The sequence representation in (2) is suitable for

GEO approach proposed in the present paper and

described in the following section.

The maintenance problem is hard to solve even

for apparently simple cases (Stapelberg, 2009), as

the time required for computing an optimal solution

increases rapidly with the size of the study case.

2.2 Generalized Extremal

Optimization for Predictive

Maintenance Scheduling

2.2.1 Extremal Optimization

The basic idea of the proposed optimization method

is inspired by (Bak and Sneppen, 1993), as a

simplified model of natural evolution in ecosystems:

a number of species in a system co-evolves to reach

the best adaptation; the worst adapted species are

forced to evolve more quickly to avoid extinction.

This mechanism determines an overall adaptation

for the entire ecosystem. In fact, according to the

Bak-Sneppen model, a macroevolutionary

ecosystem pattern is characterized by durable

periods of quiescence interrupted by some burst of

rapid considerable change. In every part of this

pattern, it is possible to observe ecological

phenomena of different size, larger ones during the

periods of major activity and smaller ones in the

more quiet periods. The size and the frequency

distributions of these events follow typically a power

law that implies 1) dynamics is unique and it

underlies both the large and the small events (scale

invariance) 2) macroevolutionary behaviour of the

global ecosystem emerges spontaneously by local

interactions between species.Both features are

considered as key issues of a working definition of a

particular state, known in statistical physics as Self-

Organized Criticality (SOC) (Turcotte, 1999), in

which system fluctuates about conditions of

marginal stability without intervention of external

factors (Bak, Tang and Wiesenfeld, 1987).

ICEC 2010 - International Conference on Evolutionary Computation

Bak-Sneppen model can be simulated through an

algorithm in quite few steps. First, for each species,

a fitness value in the range [0,1], is drawn from a

random uniform distribution. Then, the worst

adapted species (the one with least fitness) mutates

and a new fitness value is assigned to it. The change

of the worst adapted species disfigures the fitness

landscape locally, involving also the fitness of its

neighbours. For this reason, they are constrained to

mutate too, even if they are well adapted. After some

iteration, the whole system evolves toward a critical

threshold value, bringing all species to a generalized

better level of adaptation.

The model described above directly inspires the

Extremal Optimization algorithm (Boettcher and

Percus, 2001). If we set C

as a candidate solution to

an examined problem composed of |C| design

variables denoted by

its basic heuristic procedure

is the following:

1. Initialize the variables x

of C

at will; set C

best

= C.

2. For the current solution C,

a) set a fitness F

to each

variable x

b) find j such that F

is better

than F

for all i,

c) choose C’ in a neighborhood N(C)

of C so that x

must change

d) accept C = C’ unconditionally,

e) if F(C) - the total fitness of

the solution C - is better

than F(C

best

) then set C

best

= C.

3. Repeat step (2) as long as

desired.

4. Return C

best

and F(C

best

Although the above procedure shows good

performance in problems where there exist many

neighbourhood configurations of C, otherwise it

leads to a deterministic process that damages the

search of the optimal solution. To avoid this, the

algorithm was modified in some steps by

introducing into a single parameter (Boettcher and

Percus, 2001).

In particular, the steps 2.b and 2.c, have been

modified as follow: in step 2b the |C| variables x

are ranked so that to the variable with the worst

fitness is assigned rank 1, and to the one with the

best fitness rank |C|. Each time the algorithm

passes through step 2c a variable is chosen to be

mutated according to a probability distribution of the

k ranks, given by:

P(k)≈k

−

,1≤ k ≤ N,

(3)

where τ is a positive setting parameter. By means of

the parameter τ, the algorithm can choose any

variable to evolve, although the most probable

variables are those with worst fitness.

2.2.2 Generalized Extremal Optimization

and his Application

Unfortunately, EO application to a broad class of

problems is limited by some drawbacks. One of

these consists in giving a general definition for the

fitness of the single species, and this means that

different problems have different ways to assign the

fitness to the variables (Boettcher and Percus, 2001).

To clear the hurdle, De Sousa and Ramos

devised a generalization of the EO called

Generalized Extremal Optimization (GEO) (De

Sousa and Ramos, 2002) capable to tackle either

continuous, discrete or integer variables. In GEO,

the variables of the optimization problem are

arranged in a string similar to a GA chromosome, as

it is shown in Figure 2.

Figure 2: An example of the GEO encoding: N design

variables of 6 bits. Each bit is considered as a species (De

Sousa and Ramos, 2002).

This section deals with the details of a GEO

approach to the predictive maintenance scheduling

problem solving - that is also a direct way to

illustrate the GEO procedure.

The goal of the proposed method is to find the

best sequence, expressed as in (2), that minimizes

the objective function (1) for the problem described

in the section 2.1.

Let us consider a sequence (i.e., a maintenance

schedule); as aforementioned, a sequence can be

encoded in a binary string, denoted by S of length

(M*T) by means of the representation shown in (2).

This manner to express a sequence is particularly

suitable to be faced with a GEO. Indeed, in analogy

to what EO algorithm does, GEO works on a

population (configuration) by muting, generation

after generation, a single species (component) and

by estimating the obtained candidate solution, with

the aim to reach the optimal one.

A GENERALIZED EXTREMAL OPTIMIZATION-INSPIRED ALGORITHM FOR PREDICTIVE MAINTENANCE

SCHEDULING PROBLEMS

Thus, if one assumes that every representation bit

encodes a single species then an entire population

can be expressed by means of a binary string, hence

by a sequence in the form (2) too. Figure 3 illustrates

the correspondence between bit-species and

sequence-population in a visual way.

For the above reasons, a GEO algorithm can

straightly work on a sequence S by evaluating the

candidate solution to the considered maintenance

problem through the cost function (1). This means

that the lesser is the cost of the sequence the better is

the scheduling. Differently than the EO algorithm, at

each bit (species) is assigned a fitness value

proportional to the decrease of the function (1)

computed for the sequence with that bit flipped (i.e.,

mutated from 1 to 0 or vice versa).

Then, each bit is ranked, such that: to the one

with the least fitness is assigned rank 1, while to the

one with the best fitness rank N.

Subsequently, a new sequence is generated by

flipping a bit chosen according the probability law

(3) defined on the rank set.

Figure 3: A candidate solution in our GEO approach is a

sequence S (evaluated by (1)), composed of (M*T) bits, as

defined in (2). In this example, M=6 components are

maintained by n=3 resources in the time horizon T.

This iterative process halts after a prefixed

number of generation, and it returns the best

sequence S

best

which minimizes the objective

function (1).

The proposed procedure is described by the

following pseudo-code:

1. Initialize a bit sequence S

(with size M*T) randomly and

evaluate the objective function

C (as in (1));

Set: S

best

= S and C

best

= C(S);

2. For each generation:

a) For each bit i of S:

- Change the bit i (from 1 to 0,

or vice versa) and evaluate

the cost C(S

)(as in (1))for

;

- Evaluates the fitness of bit i

as ΔC(S

)=C(S

)- C

best

- Restore the bit i to its

previous value.

b) Sort ΔC(S

) in ascending way;

c) Choose the bit to change with

probability (3);

d) Set S = S

and C = C(S

);

e) If C < C

best

then set C

best

= C, and

best

= S;

3. Return S

best

and C

best

It is worth underlining that, as regard to the

traditional evolutionary algorithms (GA, SA and so

on), the present procedure has twofold advantage: (i)

there is only one adjustable parameter τ, so it

simplifies the setting task and (ii) the entire

evolution is made on one configuration solution S at

the time, unlike the traditional evolutionary

population based algorithm, and this entails lesser

computational costs and a better memory

management.

3 EXPERIMENTAL RESULTS

Preliminary experimental tests are carried out in

order to validate the effectiveness of the proposed

GEO algorithm in comparison with a standard GA

(SGA) on four high dimension scheduling cases of

the problem described in section 2.1. The problem

parameters are reported in Table 1, while the

parameter settings for both algorithms are reported

in Table 2.

Table 1: Problem parameters: the bit encoded solution

length, as defined in (2), is evaluated by means of (M*T)

bits.

Problem parameters

# M n T Solution length

(2) [bit]

1 6 2 6 36

2 8 3 8 64

3 10 4 10 100

4 10 4 15 150

ICEC 2010 - International Conference on Evolutionary Computation

As it should be noted from Table 2, the

population size for both algorithms are not reported.

This because the proposed GEO works only on 1

individual and performs a number of evaluations as

the solution length. Therefore, for example, in the

problem #1 the proposed GEO performs 36

evaluations at each iteration, while, for the problem

#3 it performs 100 evaluations at each iteration.

Therefore, in order to compare the proposed GEO

and SGA, the population size of the last algorithm

has been set to: 36 for the problem #1, 64 for the

problem #2, 100 for the problem #3 and, finally, was

150 for the problem #4.

For each problem, it was performed 50 runs for

both algorithms.

In Table 3 and Table 4 are reported the

preliminary experimental results.

In particular, Table 3 shows a comparison

between the costs of the best solutions (mean value

and standard deviation) achieved for GEO and SGA.

As one can see, both algorithms have the same

performance on the first two cases (#1 and #2), but

GEO outperforms SGA better and better while

increasing the size of the sequence.

However, in Table 4, the difference between the

algorithm presented in this work and SGA is

noticeable.

Table 2: Parameter settings for the GEO application and

the standard GA.

Proposed GEO GA

τ = 0,75

Mutation mechanism

Uniform

Crossover mechanism

Single point

Crossover fraction

0.8

Selection mechanism

Roulette

In particular, the proposed approach obtains the best

solution in lesser number of iteration on the average,

highlighting appreciable results.

Table 3: Best solution achieved (mean and standard

deviation) by means of the GEO algorithm and the

standard GA, for the 4 scheduling cases of Table 1.

Comparison test: best solution cost

Mean Standard deviation

GEO GA GEO GA

1 70 70 0 0

2 104 104 10

3 164,04 170,1 3,97 4,06

4 219,54 400,01 4,95 4,12

Table 4: Number of iteration on average and standard

deviation to achieve the best solution by means of the

GEO application and a standard GA, for the 4 scheduling

cases of Table 1.

Comparison test: number of iteration

Mean Standard deviation

GEO GA GEO GA

1 65,12 3876 60,06 2177,16

2 1804,12 3636,6 1589,75 1961,98

3 3081,18 5306,5 1979,98 2062,98

4 8513,54 16667,3 5761,65 9164,24

4 CONCLUSIONS

In the present paper, a Generalized Extremal

Optimization (GEO) based algorithm for a

predictive maintenance scheduling problem has been

proposed.

Preliminary tests on a set of high dimension

scheduling problems for the GEO algorithm

compared with a standard GA shown encouraging

performance of the proposed approach.

In particular, the proposed GEO reaches the best

solution in lesser number of iteration on average,

compared with the standard GA.

Finally, as previously mentioned, the proposed

GEO has a peculiar feature: a single candidate is

processed at each iteration.

For this reason, a comparison between an

evolutionary algorithm having the same feature (as,

for example, Simulated Annealing) and the proposed

one, should be carried out in the future research.

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ICEC 2010 - International Conference on Evolutionary Computation