Hybridized Particle Swarm Optimization for Aircraft Inspection
Check and Continuous Airworthiness Maintenance Program
Asyraf Nur Adianto and Nurhadi Siswanto
Institut Teknologi Sepuluh Nopember
Keywords: Aircraft Maintenance, Maintenance Scheduling, Particle Swarm Optimization, Greedy Randomized Adaptive
Search Procedures.
Abstract: This research compares the performances of two metaheuristic methods: Particle Swarm Optimization (PSO)
and a hybridized PSO method with Greedy Randomized Adaptive Search Procedures (GRASP) for solving
aircraft maintenance problems (AMP). In this problem, AMP consists of two different maintenance activity
types: inspection and continuous airworthiness maintenance programs (CAMP). The purpose of this paper is
to determine the number of periods that the aircraft needs to be maintained and which inspection and CAMP
tasks need to be done in each period. The problem is NP-Hard in nature, so that metaheuristic methods are
used to make sure the optimization process can be solved quickly. Computational experiments are performed
by using 16 conditions, and four randomly generated dataset instances. The computational experiment result
shows that PSO-GRASP outperforms PSO for a larger planning horizon.
1 INTRODUCTION
Aviation industries obtain their revenues based on the
number of passengers they serve by using their
aircraft (Gargiulo, Pascar, & Venticinque, 2013).
However, sometimes, the aircraft need to be
maintained. When an aircraft needs to be maintained,
it cannot be operated to earn revenues for the owners.
All airlines hope that their aircraft have high
utilization to serve their customers by minimizing the
number of maintenances for their aircraft without
violating any regulation related to aircraft
airworthiness. Research related to those problems is
termed aircraft maintenance / aeronautical
maintenance (Gargiulo et al., 2013; Han, Cao., &
Yang, 2012).
To the best of our knowledge, the research related
to aeronautical / aircraft maintenance only considered
inspection check schedule (example A-Check, C-
Check, D-Check) for the aircraft to be scheduled in
the form of aircraft maintenance routing problem
(AMRP). Several researches have been conducted in
the area of AMRP, such as: Al-Thani, Ben Ahmed,
and Haouari (2016), Eltoukhy, Chan, Chung, and Niu
(2018), Ezzinbi, Sarhani, El Afia, and Benadada
(2014) Gopalan and Talluri (1998) Liang,
Chaovalitwongse, Huang, and Johnson (2011) and
Safaei and Jardine (2018). In practice, there are other
maintenance activities that need to be considered by
the airlines, like the Continuous Airworthiness
Maintenance Program (CAMP). The difference
between inspection check and CAMP is that
inspection check does visual checking on some
components (Nickles, Him, Koenig, Gramopadhye,
& Melloy, 1999) and determines whether the
components need to be replaced or further maintained
(this maintenance activity could be categorized as
unscheduled maintenance), while CAMP does minor
maintenance based on manufacturer direction and
must be done regularly, known as scheduled
maintenance (U.S. Department of Transportation &
Federal Aviation Administration, 2016). Both
maintenance categories use maintenance resources
and could make the airlines bear the cost, but those
tasks should be done by them in order to ensure their
aircrafts’ airworthiness. These problems encourage
the airline to design a maintenance schedule that can
either reduce its costs or maximize the aircraft's'
utility.
Given the nature of aircraft maintenance, we
consider this problem based on inspection and CAMP
Adianto, A. and Siswanto, N.
Hybridized Particle Swarm Optimization for Aircraft Inspection Check and Continuous Airworthiness Maintenance Program.
DOI: 10.5220/0009403900290035
In Proceedings of the 1st International Conference on Industrial Technology (ICONIT 2019), pages 29-35
ISBN: 978-989-758-434-3
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
29
as NP-Hard. To tackle this problem, metaheuristic
methods such as Particle Swarm Optimization (PSO)
are used as an optimal solution search engine. PSO
has a broad experience in resolving maintenance
schedule. In this research, we would like to test
hybridized PSO-GRASP to find whether it could
solve the problem and outperform the PSO method.
2 PROBLEM DEFINITION
Consider a set of
inspection tasks
and a
set of
CAMP tasks
must be
scheduled on a single aircraft. Each set has notable
data, such as next do time, interval time, threshold
time, and duration (only for the set I). We consider
that in the set, and have two different time units,
such as calendar days and flight hours. Both tasks
must be scheduled on a set of
periods
which could be calculated first by using planning
horizon , Both next do
for inspection task and
for a CAMP task, and both intervals
for
inspection task and
for a CAMP task, with the
following equation (1(1).
T =
h
min
min
iI
ix
i
, min
rR
iy
r
+1
(1)
In order to use the model, available data must be
converted from two different time units into one
calendar day’s unit. For next do, interval, threshold,
and duration data of inspection tasks are converted
into the following equations (2), (4), (6), and (8),
consecutively. For next do, interval and threshold
data of CAMP tasks are converted by the following
equations (3), (5) and (7), respectively.
ncx
i,1
nx
i
nx
i
/ u
,
,
calendar days
flight hours
(2)
ncy
r,1
ny
r
ny
r
/ u
,
,
calendar days
flight hours
(3)
tcx
i
tx
i
tx
i
/ u
,
,
calendar days
flight hours
(4)
tcy
r
ty
r
ty
r
/ u
,
,
calendar days
flight hours
(5)
icx
i
ix
i
ix
i
/ u
,
,
calendar days
flight hours
(6)
icy
r
iy
r
iy
r
/ u
,
,
calendar days
flight hours
(7)
dcx
i
dx
i
dx
i
/ u
,
,
calendar days
flight hours
(8)
Those indexes and converted variables are used to
determine the maximum utilization of an aircraft by
following the equation below.
max Z =
c
t+1
- c
t
c
T+1
+m
T
(9)
x
i,t
≥ 1,
(10)
y
r,t
≥ 1,
(11)
,
,
(12)
,
,
(13)
m
t
-
x
i,t
dcx
i
= 0,
(14)
ndx
i,t
- x
i,t
c
t+1
+ icx
i
+ m
t
= 0, i I
, t T
(15)
ntx
i,t
-
1 - x
i,t
ncx
i,t
+m
t
= 0, i I
, t T
(16)
ncx
i,
t+1
–
ndx
i,t
+ntx
i,t
= 0,i I
, t T
(17)
ndy
r,t
- y
r,t
c
t+1
+icy
r
+m
t
= 0, r R
, t T
(18)
nty
r,t
- 1- y
r,t
ncy
r,t
+m
t
= 0, r R
, t T
(19)
ncy
r,
t+1
-
ndx
i,t
+ntx
i,t
= 0, r R
, t T
(20)
x
i,t
ncx
i,t
- c
t+1
≥ 0, i I
, t T
(21)
y
r,t
ncy
r,t
- c
t+1
≥ 0, r R
, t T
(22)
ncx
i,
t+1
- tcx
i
≥ 0, i I
, t T
(23)
ncy
r,
t+1
- tcy
r
≥ 0, r R
, t T
(24)
c
1
= 0
(25)
(26)
Equation (9) is the objective function of the
model. Equation (10) and (11) ensure the decision has
at least one task to be done in each period for
inspection task and CAMP task, respectively.
Equation (12) and (13) ensure generated next do data
in period always have greater or equal value
with next do data in period for inspection task and
CAMP task, respectively. Equation (14) calculates
total maintenance duration equal to all maintenance
durations of inspection task that has to be done in
period. Equation (17) and (20) calculate next do data
of both of inspection task and CAMP task,
respectively. Equation (17) and (20) could be
calculated by equation (15) and (18) for calculating
the difference value when the task is being done in
period, while equation (16) and (19) for calculating
the difference value when the task is not being done
in period. Equation (21) and (22) ensure next do data
of period always have greater or equal value with
current days of period for inspection task and
CAMP task, respectively. Equation (23) and (24)
ensure next do data for period always have
greater or equal to threshold data for inspection task
and CAMP task, respectively. Equation (25) ensures
the current days of period 1 have a value equal to 0.
Equation (26) ensures the decision variable for both
ICONIT 2019 - International Conference on Industrial Technology
30
the inspection task, and the CAMP task is filled with
binary value.
3 MODEL DEVELOPMENT
In this research, we focused on solving the given
problem using two metaheuristic methods, pure PSO
and hybrid PSO-GRASP. In this following sub-
section, the PSO and GRASP will be first presented,
and then their combination to create the PSO-GRASP
is discussed.
3.1 Particle Swarm Optimization
PSO is a swarm intelligent metaheuristic method
based on how a flock of birds tracks down their prey
(Santosa & Ai, 2017). Every bird will fly to the best
location based on all information shared by other
birds in that flock, including itself. To reach the best
objective function for each iteration, each individual
has to share their objective function value and
conclude all information into one best. In order to
make an individual move to its targeted value, the
individual velocity needs to be updated as in the
following equations (27) and (28).
v
p,
a+1
=ωv
p,a
+b
1
r
1
pbest
p,a
-x
p,a
+b
2
r
2
gbest
a
-x
p,a
(27)
(28)
Both equations consist of the following index:
▪ Set
individual or population
1, 2, …, P
▪ Set A
iteration
1, 2, …, I
Both equations use the following related
variables:
▪ v
p,
a: velocity of p population of an iteration
▪
a: generated solution in p population of an
iteration
▪ pbest
p,
a: best-generated solution of p
population in an iteration
▪ gbest
a
: best-generated solution of all
population in an iteration
▪ r
2
: randomized number with
decimal value from 0 to 1
In equation (27), there are some parameters that
can be set up manually, which are:
▪ b
1
: ratio to take the best-generated
solution of each population that would affect
the velocity variable
▪ b
2
: ratio to take the best-generated
solution of all population that would affect
velocity variable
▪ ω : ratio to take an earlier iteration of
generated solution that would affect velocity
variable
In equation (27), we will calculate velocity of
iteration by determining a certain portion of
velocity in an earlier iteration, a certain portion of the
gap between the generated solution of the last
iteration with best generated solution of p population,
and a certain portion of the gap between the generated
solution of the last iteration with the best generated
solution of all population. Using those velocity
values, we can update the new position of each
individual using equation (28) by adding the
generated solution of the last iteration with calculated
velocity from equation (27).
3.2 Greedy Randomized Adaptive
Search Procedures (GRASP)
GRASP is a metaheuristic method designed for
helping other metaheuristic methods to find an
optimal solution efficiently. This method was
developed by Feo dan Resende (1995) and had two
phases, the construction phase, and the local search
phase. Both phases are developed by specifically
identifying the decision variable of the problem and
constructing the encoded form of those decision
variables.
Table 1: Decision Variable Transformed Form
Per 1
Per 2
...
Per T
1
0
...
1
0
1
...
1
...
...
...
...
0
0
...
0
0
0
...
1
1
1
...
1
...
...
...
...
1
0
...
1
In this research, decision variables of the problem
have two-dimensional matrix;
, and
which
represents each inspection and CAMP task,
respectively, at each period. Metaheuristic methods
must generate those variables, as illustrated in Table
1. Using this form, the metaheuristic method
sometimes may generate a solution that violates some
constraints of the problem. To avoid generating an
invalid solution, GRASP is implemented at both
metaheuristic methods and creates a new decision
Hybridized Particle Swarm Optimization for Aircraft Inspection Check and Continuous Airworthiness Maintenance Program
31
variable form filled with inspection task ID, as
illustrated in Table 2.
Table 2: Decision Variable with GRASP Transformed
Form
Per 1
Per 2
...
Per T
3
4
...
2
3.3 Hybrid PSO-GRASP
PSO method must generate data by filling the form as
described in the previous section with the decimal
value from 0 to 1, as illustrated in Table 3. Those
decimal values are converted into inspection task ID
by finding the nearest multiplication of decimal value
with . For example, if , the generated form in
Table 3 will be transformed, as illustrated in Table 2.
Table 3: Decision Variable with GRASP Form
Per 1
Per 2
...
Per T
In-ref
0.439
0.72
...
0.272
To convert generated decision variables with
GRASP, as in Table 2, into the original decision
variable form as in Table 3, we use the local search
method, which satisfies equation (29) and (30). Both
equations ensure the selected inspection and CAMP
tasks that have to be done next must-have lower or are
equal to the referred inspection task’s next values.
ncx
ref
t
,t
-
x
i,t
ncx
i,t
≥0
(29)
ncx
ref
t
,t
-
y
r,t
ncy
r,t
≥0
(30)
3.4 Proposed Model
The proposed algorithm of Hybrid PSO-GRASP is
depicted in Figure 1. First, we generate the referred
inspection task using a random number generator and
convert them using the GRASP method for the first
iteration, using PSO-GRASP later.
For the iteration, until the maximum iteration
number is reached, we update the max time of each
maintenance period in each generated solution, then
construct all inspection and CAMP tasks that need to
be ordered. Using the ordered task, we compute the
maintenance duration needed. Using the ordered task
and the maintenance duration in each period, we
calculate when inspection and CAMP tasks should be
done and then update the utilization of the aircraft.
Using the ordered tasks, we validate the generated
solution each period. Each valid solution will be
marked and will be used to calculate the objective
function value of the generated solution.
The objective function value is calculated from
utilization value for a valid solution in each period
and penalty value for an invalid solution in each
period. Each period’s penalty value has a different
number. Increased period value means decreased
penalty value, and the highest penalty value will be in
the first period. Penalty value will be calculated
Generate referred inspection tasks.
For ( from p ← 1 to p ← P
)
For ( from t ← 1 to t ←T
)
Update the max period of time.
For ( from i ← 1 to i ← I
)
Construct task order of all inspection tasks.
End
For ( from r ← 1 to r ← R
)
Construct task order of all CAMP tasks.
End
Update maintenance duration.
Update next do value of the next period.
Update utilization of the aircraft.
Validate the generated solution.
End
Compute the objective function value for each
solution population.
End
Save the maximum objective value of each generated
population
Figure 1: Pseudocode of PSO-GRASP Model
Following equation (31). When we have a
maximum period T
equal to 30 and the generated
solution has an invalid solution at a few period t =
1, 3, 15
, then we have penalty
1
equal to 3000,
penalty
3
equal to 2800, penalty
15
equal to 1600. From
those values, we can conclude that the generated
solution has a total penalty equal to 7400.
(31)
The objective function value is calculated using
calculated utilization and penalty value following
equation (32). When the generated solution in p
population calculates its objective function, the
current value of the objective function will be added
by summation of utilization time in each period that
is marked as valid solution divided by next do time at
last period T
added by maintenance duration of last
period T
as described in equation (9). When the
generated solution in p population and t period are
marked as invalid solution, the current value of the
objective function will be subtracted by penalty
value, as described in equation (31).
ICONIT 2019 - International Conference on Industrial Technology
32
Table 4: The Results by Varying Planning Horizon and Dataset Size Sensitivity Testing
ID
Plan
Horizon
Group
PSO
PSO-GRASP
Last Maint.
Finish Time
Obj Value
CPU
Time
Last Maint.
Finish Time
Obj
Value
CPU
Time
Test01
730
G1
453
-277
600.39
782
78.14
110.45
Test02
G2
267
-3650
600.64
781
78.11
181.19
Test03
G3
1008
-11680
600.25
1100
49
258.64
Test04
G4
807
-20440
600.15
923
48.43
390.59
Test05
1460
G1
639
-821
600.69
1776
77.03
197.65
Test06
G2
480
-2920
600.19
1673
77.35
337.11
Test07
G3
1839
-154760
600.85
2361
46.3
508.42
Test08
G4
1804
-143080
600.93
2362
43.4
744.06
Test09
2190
G1
856
-26280
601.53
2819
76.84
289.77
Test10
G2
759
-56940
600.53
2669
76.89
496.17
Test11
G3
2636
-575970
600.15
4171
41.99
600.32
Test12
G4
2528
-917610
600.93
4419
41.46
600.52
of
value
p
-
vld
p,t
c
T+1
+m
T
+
penalty
t
1 vld
p,t
(32)
4 EXPERIMENTAL DESIGN
This research evaluates the performance of both the
non-hybridized PSO method and PSO that have been
hybridized with GRASP with four sets of randomized
problems’ data. These instances are generated on a
different scale, from small scale to large scale, with
five and 20 inspection tasks, combined with 500 and
1000 CAMP tasks. The combination of each dataset
is described in Table 5. For example, Group G1 has a
combination of five inspection tasks and 500 CAMP
tasks. Moreover, PSO has three parameters that could
be set up manually, consisting of
,
, and . In this
experiment, we use the basic values of PSO
parameters, which are
.
Table 5: Dataset Instance
Group
Inspection Task
CAMP Task
G1
5
500
G2
5
1000
G3
20
500
G4
20
1000
5 COMPUTATIONAL RESULT
Both PSO and PSO-GRASP were executed until the
objective value for the generated solution was labeled
as a valid solution and reached the maximum iteration
parameter. The other set condition was that the
objective value is invalid, and the iteration process is
stopped when CPU computational time is greater than
or equal to 600 seconds. Using the parameters that
have been set up before, we do two kinds of
experiments. The first experiment is to do sensitivity
testing on both varying planning horizon parameters
and the size of the dataset. The second experiment is
to do sensitivity testing on both iteration and
population parameters.
The first experimental results are shown in Table
4. Increasing the planning horizon of the test has an
impact on increasing CPU time in both PSO and PSO-
GRASP methods. In the example for dataset G1 using
PSO-GRASP, for planning horizon 730 days has
110.45 seconds of computational time, while for
planning horizon 1460 days, the CPU time increases
to 197.65 seconds, and for planning horizon, 2190
days CPU time usage increased to 289.77. For the
other method, PSO used at least 600 seconds CPU
time in all conditions. PSO always generates invalid
solutions because, in every test, the objective values
of PSO always have negative values, as we have
explained in sub-section 3.4 so that the values are
invalid solutions.
Experiments by using PSO-GRASP on the larger
size as well as longer planning horizon datasets use
more CPU time, either in inspection task number or
CAMP task number. From all tests in this first
experiment, 83.3% of the results show that PSO-
GRASP uses lower CPU time compared with the
PSO, while the other 16.6% have the same average
CPU time as PSO.
Increasing the planning horizon of the tests can
decrease objective function values. In the example for
dataset G1 using PSO-GRASP, 730 days of planning
horizon has an objective function value Equal to
78.14, 1460 days has 77.03, and 2190 days have
Hybridized Particle Swarm Optimization for Aircraft Inspection Check and Continuous Airworthiness Maintenance Program
33
76.84. The same condition also applies for a larger
size dataset, which mostly generates less objective
function value, either in inspection task number or
CAMP task number, both using PSO-GRASP. For
1460-day planning horizon, using PSO-GRASP,
dataset G1 until G4 has objective function values
equal to 77.03, 77.35, 46.3, and 43.4, consecutively.
Based on these results, it can be concluded that
increasing the number of CAMP tasks does not
significantly change the objective function values
Table 6: The Results by Varying the Number of Iteration and Population Sensitivity Testing
ID
Iteration /
Population
Limitation
Group
PSO
PSO-GRASP
Last
Maint.
Finish
Time
Obj
Value
% Gap
CPU
Time
Last
Maint.
Finish
Time
Obj
Value
% Gap
CPU
Time
Test13
1000/100
G3
425
-305
-
600.44
1051
47.48
-
240.68
Test14
1000/200
467
-263
13.77%
600.32
861
53.08
11.79
475.16
Test15
2000/100
474
-256
16.07%
600.59
1146
49.92
5.14
481.61
Test16
2000/200
426
-304
0.33%
600.45
914
54.6
15.00
930.88
The second experiment explored only Dataset G3
with 730 days planning horizon, as shown in Table 6,
by varying the number of iterations and population
Test13 uses 1000 iterations and 100 populations as
the baseline. Test 14 uses the same number of
iterations as the one in Test 13; however, the number
of populations of Test 14 is double the one of Test 13.
Test 15 and Test 16 have the same number of
iterations, 2000, but they differ in the number
population.
Increasing the number of population and the
number of iteration parameters escalates the CPU
time and mostly increases the objective function
values in both methods. In PSO-GRASP, doubling
the number of populations significantly increased the
objective function values with a 15% gap rather than
doubling the number of iteration parameters with only
a 5.14% gap.
Using PSO, increasing the number of iteration
parameters significantly increases the objective
function value with a 16.07% gap rather than that
increasing the population parameter with a 13.77%
gap and that increasing both iteration and population
parameters with 0.33% gap. Even though the number
of iterations and population parameters may affect
PSO performance, we cannot make any accurate
conclusion of CPU time usage because the objective
function generated by the PSO method is always
marked as an invalid solution and cannot be
implemented to the real system.
6 CONCLUSIONS
In this paper, we presented a MILP model for
optimizing aircraft maintenance scheduling
problems, considering the inspection check task and
CAMP task. Both tasks have to be considered by the
airline because it could affect the airworthiness of
their aircraft. Because of the NP-Hard nature of the
problem, we developed PSO-GRASP metaheuristic
methods to solve this problem in a reasonable time.
We tested the model using four randomly generated
datasets. We compared the metaheuristic models,
PSO, and hybrid PSO-GRASP, based on objective
function values and their CPU computational times.
The developed model could solve both small and
large-scale datasets. Using a larger scale dataset, the
result showed that the model could generate small
objective function value, but it needs longer CPU
time when tested in a similar parameter setting.
Statistical analysis shows that the PSO-GRASP
model is able to provide better performance than the
PSO method without hybridization based on the
objective function values. PSO cannot even provide a
valid solution to this problem.
By using the larger values of the iterative and the
population parameters, it makes PSO-GRASP work
better, but, in the PSO, the changes in these
parameters do not have any impacts on the solutions
made and still provide an invalid solution. Moreover,
by using these parameters or other parameters such as
the planning horizon, the computational times are
longer both for PSO and PSO-GRASP.
An interesting topic following this research is to
develop the exact algorithm, whether using Mixed
Integer Programming or Non-Linear Programming,
using either small or larger similar datasets.
Developing other metaheuristic methods, such as the
Genetic Algorithm, Tabu Search, or Simulated
Annealing, is also interesting for the next research
agenda for solving this aircraft maintenance problem.
ICONIT 2019 - International Conference on Industrial Technology
34
ACKNOWLEDGMENTS
The authors wish to acknowledge the funding of this
research is from the Ministry of Research Technology
and Higher Education, the Republic of Indonesia,
through the Postgraduate Thesis Research Scheme
Grant No. 781/PKS/ITS/2019.
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