An outclassing Multi-objective Hybrid Genetic-based Discrete PSO for
Solving the PECT Problem
DOME LOHPETCH
Department of Mathematics, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800,
THAILAND
Abstract: - The Post Enrolment based Course Timetabling (PECT) Problem belongs to, one of the classical
problems, the timetabling problems, and it is a part of the most real-life problems that come with multiple
constraints of nature. Such a problem is investigated together with both hard and soft constraints, and the
solution is an optimal timetable satisfying both constraints as far as possible which reflects the quality of the
solution. As a result, there are many approaches to solving the PECT Problem. However most approaches rely
upon both the determination of parameters or understanding of domain knowledge. In this research, the
Genetic-based Discrete Particle Swarm Optimization (PSO) has been developed with two different local search
approaches: Local Search and Tabu Search to solve multi-objective functions and get good solutions by
improving the performance of searching solution, which has few parameters to be tuned, and it can outperform
all related algorithms from the published work.
Key-Words: - multi-objective optimization problem, hybrid algorithm, genetic-based discrete particle swarm
optimization, local search, tabu search, post enrolment based course timetabling problem.
Received: May 16, 2024. Revised: October 17, 2024. Accepted: November 19, 2024. Published: December 30. 2024.
1 Introduction
The Post Enrolment Based Course Timetabling
Problem (PECTP) is a real-world problem, which is
a problem that occurs continuously in all
universities. However, the resources and constraints
of each university are different from one university
to another. It is an NP-complete problem, [1], which
is well known that there is no available algorithm to
solve with the degree of polynomial running time.
Moreover, the PECTP is related to resource
allocation such as events, features, and students into
optimal rooms and timeslots that are scheduled
using the completed enrolments of all students,
classified as a combinatorial optimization problem,
[2]. In this work, the representation of a solution
with discrete variables was designed to suit the
nature of the problem described above, and it also
helps to quickly seek a feasible solution, resulting in
a more effective solution. In addition to designing
the representation of a solution, constraints are the
main important factor to consider carefully, and they
are classified into hard constraints and soft
constraints. In case of violating hard constraints, the
solution is infeasible. However, the soft constraints
can be violated, and the quality of the solution is
measured by the number of soft constraint
violations. As a result, the soft constraints must be
satisfied as much as possible, and it also directly
affects the efficiency of the solution.
The meta-heuristic algorithms will be used as
the main tool emphasized on nature-inspired
algorithms in this research, and there are many
popular meta-heuristic algorithms such as Genetic
Algorithm (GA), Particle Swarm Optimization
(PSO), Ant Colony Optimization (ACO), Local
Search (LS), Tabu Search (TS), Simulated
Annealing (SA) and Hybrid Approaches (HA).
Particularly, Genetic-based Discrete Particle Swarm
Optimization (GDPSO) was chosen as the main
algorithm, and a single-objective algorithm to solve
PECTP that hybridizes Genetic-based Discrete
Particle Swarm Optimization with Local Search and
Tabu Search (HGDPSOLTS) was proposed in [3],
and it was able to solve the PECTP with a good
performance. However, in the real world, PECTP is
considered as multi-objective optimization problem
rather than a single-objective optimization problem.
For this reason, this research extends the
HGDPSOLTS in [3] which used the model of
single-objective function to solve the problem in the
form of a multi-objective function to see its
performance compared with [4] and the other
methods from the literature by using 11 standard
testing benchmark datasets from Metaheuristic
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Network (MN). This will provide supporting
evidence to answer the research question: Is multi-
objective model solving better than single-objective
model solving PECTP?
2 Multi-Objective PECTP
The model of PECTP used in this work was
proposed in [5], and optimum rooms and time slots
are assigned to each event based on the enrolment
data of the students with an attempt of satisfying
both hard and soft constraints.
The PECTP model consists of a set of events
, a set of 45 timeslots
(5 days of 9 hours on each
day), a set of rooms
with
different size-seating capacity on each room (events
can occur in fitting rooms), a set of students
attending each event, and a set of
features
such as computer, and
internet connection, providing by each room and
requesting by each event. The timetable framework
used in this work with x-axis representing rooms
and y-axis representing time slots is shown in Figure
1.
Fig. 1: The framework of assigning timeslots and
rooms to events of PECTP
There are three hard constraints that a feasible
solution cannot violate as follows.
H1: Students can attend only one event at the
same timeslot.
H2: All attending students can fit in the room
satisfying all required features of the event.
H3: Each room is assigned only one event in
any timeslot.
Concurrently, there are three soft constraints
that a feasible solution should satisfy as much as
possible:
S1: The event with the last time slot of a day
(timeslot 9, 18, 27, 36 or 45) should not be
attended by students.
S2: On the same day, students should not
participate in more than two events of
consecutive time slots.
S3: In a day, students should not attend only one
event.
The three soft constraints are respectively
divided into three objective functions:
and
, and they should be minimized as much
as possible. Consequently, this leads to solving the
PECTP in the form of a multi-objective function.
3 Proposed Hybrid Approach
This work proposed a hybrid multi-objective
approach, GDPSO, combined with two different
local search approaches: Local Search and Tabu
Search to solve the multi-objective PECTP, denoted
as HMGDPSOLTS. The process of HMGDPSOLTS
is different from [3] which solves the single-
objective PECTP. The pseudo-code of
HMGDPSOLTS is provided in Figure 2.
Proposed Hybrid Approach - HMGDPSOLTS
input: A problem case
1: Initializing No_of_Generation to 0
2: Creating an archive set for keeping non-dominated
solutions
3: Generating the first swarm of the solutions
4: for every solution in the first swarm do
5: Applying local search to each solution
6: Computing objective value of each objective function
of each solution
7: end for
8: Computing crowding distances and ranks of all solutions
in the first swarm
9: Placing non-dominated solutions into the archive set
10: while not meet the termination condition do
11: for Every solution in the swarm do
12: Preserving as the personal best of each
solution
13: end for
14: Preserving as the global best of the swarm
15: for every solution in the swarm do
16: Applying crossover process to and each
solution
17: Applying crossover process to and each
solution
18: Applying mutation process with the probability of
mutation at to each solution
19: Applying local search process to each solution
20: Applying tabu search process to each solution
21: Computing objective value of each objective
function of each solution
22: end for
23: Computing crowding distance and rank of all
solutions in the swarm
and the archive set
24: Placing non-dominated solutions into the archive set
25: Setting No_of_Generation increasing by one
26: end while
output: Yielding non-dominated solutions in the archive set
Fig. 2: The pseudo code of HMGDPSOLTS
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3.1 Multi-Objective Genetic-based Discrete
Particle Swarm
The first key part in the proposed hybrid method is
the multi-objective genetic-based discrete particle
swarm optimization to solve the multi-objective
PECTP, and the process is different from the
original GDPSO that solves the PECTP in terms of
single-objective function. The implementation of the
multi-objective has the property known as Pareto
dominance, [4], [6], [7], [8]. This property is used to
compare two solutions. A solution is indicated to
be Pareto optimal when there is no other solution
in search space such that dominates .
In this case, it also says that is non-dominated
concerning and keeping in the archive set. The
archive set is formed to keep the distinctly non-
dominated solutions. Finally, the non-dominated
solution is selected from the archive set.
3.2 Local Search Approach
Local search (LS) is a heuristic approach for solving
difficult optimization problems as it helps to reduce
the exploring of the search space, including quickly
seeking the feasible solution. This work used LS
from [4] as a basic framework, displayed in Figure
3.
Local Search Approach
input: Solution from the current swarm
1: Generating a sorted list of events
2: if the current solution is infeasible then
3: Investigating the first item of the sorted list
4: while not meet the termination condition and
5: the current event still has an untried move left do
6: Computing the moves of by assigning each time
slot to the current event
7: if moving leads to decreasing of the number of hard
constraint violations then
8: Moving to next event and going to line no. 4
9: end if
10: end while
11: else {the current solution is feasible}
12: Investigating the first item of the sorted list
13: while not meet termination condition and
14: the current event still has an untried move left do
15: Computing the moves of by assigning each time
slot to the current event
16: if moving leads to decreasing of the number of soft
constraint
violations with no hard constraint violations then
17: Moving to next event and going to line no. 13
18: end if
19: end while
20: end if
output: Yielding an improved solution
Fig. 3: The pseudo code of a local search approach
3.3 Tabu Search Approach
Tabu Search (TS) takes a probable solution to a
problem and investigates its immediate neighbor
solutions. TS is applied after the process of LS is
completed to further improve the effectiveness of
the solution, and this work used TS based on the TS
approach from [4] as shown in Figure 4.
Tabu Search Approach
input: Solution s from the current swarm
1: Assigning solution s as the best solution
2: Building a tabu list
3: Initializing to 0
4: while not meet the termination condition do
5: Initializing to 0
6: while 10% of all solutions do
7: Assigning as the i-th move of solution s
8: Computing objective values of all objective
functions of solution
9: end while
10: if there is some solution that has been dominated by
the solution and the solution
dominate or not dominate the solution then
11: Assigning the solution to be the solution s
12: Assigning the item at index of tabu list equal to
13: Increasing by one
14: else
15: Assigning the best solution among all ,which is
not in the tabu list list yet, to solution s
16: Assigning the item at index of the tabu list equal to
17: end if
18: Assigning the best solution so far to be the solution s
19: end while
output: Yielding the solution s
Fig. 4: The pseudo-code of a Tabu search algorithm.
4 Experiments
The experimental results of a proposed hybrid
method to solve the multi-objective PECTP on
Metaheuristic Network (MN) datasets, known as a
standard testing benchmark, are reported in this
section. These datasets can be divided into 3
categories: 5 easy classes, 5 medium classes, and 1
hard class. Moreover, the proposed approach was
compared with the related algorithms, and the
number of evaluations was used to accomplish the
fairness of comparison with other algorithms. The
nature of the three classes is specified in Table 1.
The parameters, according to [3] and [4] have
been tuned up for each class differently. The
terminating condition was the number of
evaluations, specifying as 20,000 for easy cases,
10,000 for medium case 1 to 4, 14,000 for medium
case 5 and 30,000 for hard case. For each problem
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case, all experiments were run repeatedly 20 times.
In case of GDPSO, the swarm size, , was set to 10,
while the mutation probability, , was set to 0.1.
What is more, the time limit and the
number of maximum steps were used to stop
the process of local search as follows. was
fixed to 100 for easy cases, 1,000 for medium cases
and 10,000 for hard case, respectively, and
was specified to 200 for easy cases, 1,000 for
medium cases and 2,000 for hard case, respectively.
Finally, the number of maximum iterations, ,
was set as the termination condition of tabu search,
setting to 20 for easy cases, 30 for medium cases,
and 80 for hard case.
Table 1. The Characteristics for each category of
MN datasets
Characteristic
Small
Class
Medium
Class
Hard
Class
The number of
events
100
400
400
The number of
rooms
5
10
10
The number of
features
5
5
10
The number of
students
80
200
400
4.1 Comparison with the Related
Algorithms on Multi-Objective PECTP
A multi-objective Genetic-based Discrete Particle
Swarm Optimization (GDPSO) and a hybrid multi-
objective Genetic-based Discrete Particle Swarm
Optimization with a LS algorithm (HMGDPSOLS)
have been coded in Python and compared with the
proposed approach to provide a clearer overview of
the comparison. The experiments of those
algorithms were run on the same group of machines
to replicate the same testing environment, resulting
in a fair comparison of results in each algorithm,
including the parameter as discussed above. The
abbreviations and descriptions of all 5 compared
algorithms are described in Table 2.
Table 3 and Table 4 report the comparison results
of proposed HMGDPSOLTS with all related
algorithms on easy, medium, and hard cases
respectively in terms of the average values of the
number of soft constraint violations in each
objective function. The best solution for all datasets
is highlighted, and the proposed hybrid approach
acquired more beneficial results than all related
algorithms in all cases except Easy5 instance, which
got the same average value with HNSGA2LTS as
they both were not violated all constraints resulting
in the average value of three objective functions
equal to zero. To sum up, the comparison results
clearly pointed out that the proposed method was
able to outperform all other algorithms.
Table 2. The abbreviations and descriptions of the
related algorithms on multi-objective PECTP
Abbreviations
Description
HMGDPSOLTS
The proposed hybrid approach in
this paper.
MGDPSO
A multi-objective Genetic-based
Discrete Particle Swarm
Optimization.
HMGDPSOLS
A hybrid multi-objective Genetic-
based Discrete Particle Swarm
Optimization with a LS algorithm.
HNSGA2LTS
A hybrid multi-objective genetic
algorithm with a new local search
approach for solving the post-
enrolment-based course timetabling
problem by [4] in 2016.
GSNSGA
Guided search non-dominated
sorting genetic algorithm by [9] in
2011.
Table 3. Comparison of HGDPSOLTS with the
related algorithms on the multi-objective PECTP
running on easy cases
Dataset
Algorithms
Average
f1
f2
f3
Easy01
HMGDPSOLTS
0
0
0
MGDPSO
0
0
3.25
HMGDPSOLS
0
0
0.20
HNSGA2LTS
0
0
0.08
GSNSGA
1.33
6.74
9.91
Easy02
HMGDPSOLTS
0
0
0
MGDPSO
0
0
6.40
HMGDPSOLS
0
0
1.65
HNSGA2LTS
0
0
0.64
GSNSGA
1.63
5.75
5.90
Easy03
HMGDPSOLTS
0
0
0
MGDPSO
0
0
2.90
HMGDPSOLS
0
0
1.75
HNSGA2LTS
0
0
0.20
GSNSGA
0.65
2.06
7.38
Easy04
HMGDPSOLTS
0
0
0
MGDPSO
0
0
3.50
HMGDPSOLS
0
0
2.40
HNSGA2LTS
0
0
0.20
GSNSGA
1.02
1.14
20.46
Easy05
HMGDPSOLTS
0
0
0
MGDPSO
0
0
3.05
HMGDPSOLS
0
0
2.00
HNSGA2LTS
0
0
0
GSNSGA
1.52
2.04
15.10
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Table 4. Comparison of HGDPSOLTS with the
related algorithms on the multi-objective PECTP
running on medium and hard cases
Dataset
Method
Average
f1
f2
f3
Medium01
HMGDPSOLTS
0
0
0
MGDPSO
8.05
11.85
0.10
HMGDPSOLS
3.55
6.10
0
HNSGA2LTS
72.47
110.90
5.82
GSNSGA
8.74
138.76
32.52
Medium02
HMGDPSOLTS
0
0
0
MGDPSO
2.10
3.00
0.05
HMGDPSOLS
1.25
1.05
0.15
HNSGA2LTS
74.47
111.49
5.31
GSNSGA
13.00
176.60
21.00
Medium03
HMGDPSOLTS
0
0
0
MGDPSO
2.45
1.75
0.10
HMGDPSOLS
1.85
1.05
0.25
HNSGA2LTS
106.62
135.28
6.97
GSNSGA
6.90
145.81
16.69
Medium04
HMGDPSOLTS
0
0
0
MGDPSO
6.05
2.85
0.05
HMGDPSOLS
0.35
0.95
0
HNSGA2LTS
59.74
101.28
5.30
GSNSGA
7.15
88.81
22.38
Medium05
HMGDPSOLTS
0
0
0.20
MGDPSO
0.65
0.20
5.75
HMGDPSOLS
0
0.20
6.35
HNSGA2LTS
124.47
108.72
8.95
GSNSGA
23.15
150.40
43.15
Hard01
HMGDPSOLTS
0
0
51.75
MGDPSO
0
0
115.50
HMGDPSOLS
0
0
95.85
HNSGA2LTS
587.97
446.32
37.95
GSNSGA
39.72
345.94
124.64
4.2 Comparison with the Related
Algorithms on single-Objective PECTP
To provide a more complete comparison, the
proposed HMGDPSOLTS approach was compared
with other algorithms from the published work on
single-objective PECTP as most previous works of
PECTP were in the form of single-objective
optimization. The results of HMGDPSOLTS were
transferred into single-objective experimental results
by combining three objective values into a single
objective value, and the abbreviations and
descriptions of all 19 compared algorithms are in
Table 5.
Table 6 offers the comparison results of the
proposed HMGDPSOLTS with the 18 algorithms
from the literature review in form of single-
objective PECTP on easy cases. The comparison
results revealed that the proposed HMGDPSOLTS
approach gave a better result than RRLS, FMHO,
GHH, and HAS in all easy cases. The rest of the
related algorithms had no difference in terms of
performance because they did not violate all
constraints, resulting in the value of three objective
functions equal to zero. Having said that, it can
illustrate the considerable difference in medium and
hard cases.
Table 5. The abbreviations and descriptions of the
related algorithms on single-objective PECTP
Abbreviations
Description
HMGDPSOLTS
The proposed hybrid approach in this
paper.
HGDPSOLTS
An Outperforming Hybrid Discrete
Particle Swarm Optimization for Solving
the Timetabling Problem by [3] in 2019.
HGALTS
Hybrid genetic algorithm with local
search and tabu search approach by [10]
in 2015.
MHSA
Modified harmony search algorithm by
[11] in 2012.
HAS
Harmony search algorithm by [11] in
2012.
GSGA
Guided search genetic algorithm by [12]
in 2009.
HGHH1
Hybrid graph-based hyper-heuristic with
local search on complete solution by [13]
in 2009.
HGHH2
Hybrid graph-based hyper-heuristic with
local search during solution construct by
[13] in 2009.
MA
Memetic algorithm by [14] in 2008.
GAWLS
Genetic algorithm with local search by
[15] in 2008.
HEA
Hybrid evolutionary approach by [16] in
2007.
GHH
Graph-based hyper-heuristic by [17] in
2007.
RII
Random iterative improvement by [18]
in 2007.
VNS
Variable neighborhood search by [19] in
2005.
FMHO
Fuzzy multiple heuristic ordering by [20]
in 2005.
TSHH
Tabu-search hyper heuristic by [21] in
2003.
EALS
Evolutionary algorithm with local search
by [22] in 2002.
MMAS
Max-min ant system by [5] in 2002.
RRLS
Random restart local search by [5] in
2002.
The experimental results in Table 7 have pointed
out that the proposed hybrid approach was able to
outperform 17 other algorithms on all medium
cases, while it took the same performance as
HGDPSOLTS as both showed no soft violation in
all medium cases. For a hard instance, the number
of soft constraint violations of the proposed
HMGDPSOLTS was equal to 23, and it still can
beat all 18 related algorithms, which “x%Ifea” is the
percentage of runs that cannot seek a feasible
solution (infeasible solution).
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Table 6. Comparison of HGDPSOLTS with the
related algorithms on the single-objective PECTP
running on easy cases
Algorithm
Easy0
1
Easy0
2
Easy0
3
Easy0
4
Easy0
5
HMGDPSOLTS
0
0
0
0
0
HGDPSOLTS
0
0
0
0
0
HGALTS
0
0
0
0
0
MHSA
0
0
0
0
0
HAS
3
4
2
3
1
GSGA
0
0
0
0
0
HGHH1
2
2
1
1
0
HGHH2
0
0
0
0
0
MA
0
0
0
0
0
GAWLS
2
4
2
0
4
HEA
0
0
0
0
0
GHH
6
7
3
3
4
RII
0
0
0
0
0
VNS
0
0
0
0
0
FMHO
10
9
7
17
7
TSHH
1
2
0
1
0
EALS
0
3
0
0
0
MMAS
1
3
1
1
0
RRLS
8
11
8
7
5
Table 7. Comparison of HGDPSOLTS with the
related algorithms on the single-objective PECTP
running on medium and hard cases
Algorithm
Medium
01
Medium
02
Medium
03
Medium
04
Medium
05
Hard01
HMGDPSO
LTS
0
0
0
0
0
23
HGDPSOLT
S
0
0
0
0
0
25
HGALTS
137
132
194
114
160
789
MHSA
168
160
176
144
71
417
HAS
296
236
255
231
207
-
GSGA
240
160
242
158
124
801
HGHH1
310
419
332
324
162
80%
Ifea
HGHH2
257
259
192
235
112
80%
Ifea
MA
227
180
235
142
200
-
GAWLS
254
258
251
321
276
1027
HEA
221
147
246
165
135
529
GHH
372
419
359
348
171
1068
RII
242
161
265
181
151
100% Ifea
VNS
317
313
357
247
292
100% Ifea
FMHO
243
325
249
285
132
1138
TSHH
146
173
267
169
303
80%
Ifea
EALS
280
188
249
247
232
100% Ifea
MMAS
195
184
248
164.
5
219.
5
851.5
RRLS
199
202.
5
77.5
%If
ea
177.
5
100
%If
ea
100%Ifea
Furthermore, it is noted that 9 comparator
algorithms were unable to seek a feasible solution
on a hard instance. To conclude, the proposed
hybrid approach, HMGDPSOLTS, has shown the
outperformance of the experimental results in all
cases compared with all related algorithms on both
the single-objective and multi-objective functions.
5 Conclusion
To sum up, the proposed hybrid algorithm has
mainly been tested for its performance in solving the
multi-objective PECTP on the standard testing
benchmark, MN datasets. A hybrid approach
integrates local search and tabu search techniques
into a genetic-based discrete particle swarm
optimization, denoted HMGDPSOLTS. Overall, the
empirical results showed that the proposed hybrid
method for solving the single-objective and multi-
objective PECTP gave an outstanding performance,
and it was able to get the best feasible solution of no
soft constraint violation, resulting in the value of
three objective functions equal to zero. It
outperforms all published work on the same testing
benchmark. What is more, the satisfactory
performance of the proposed hybrid approach arises
from the two parts. The first one is the designing of
the solution representation with discrete variables,
and this gives rise to a marked effect on the
performance in terms of quickly seeking a feasible
solution more effectively. The second part is the
appropriate hybridization of LS and TS algorithms,
and this brings about the ability to seek the neighbor
solutions of a local search and a tabu search in the
forms of exploitation and avoidance from the local
optima.
In future work, the aims are to test the proposed
hybrid method to solve the PECTP in terms of both
single-objective and multi-objective optimization
problems on real-world datasets such as the datasets
from the real-life university in Thailand and to
consider other soft constraints such as the utilization
of rooms to minimize the cost of teaching and
learning in the university.
Acknowledgement:
This research has been supported for machines to
run the experiments by the Department of
Mathematics, Faculty of Applied Science, King
Mongkut’s University of Technology North
Bangkok.
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