A Mathematical Programming Model and Enhanced Simulated Annealing Algorithm for the School Timetabling Problem
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Asian Journal of Research in Computer Science
5(3): 21-38, 2020; Article no.AJRCOS.55744
ISSN: 2581-8260
A Mathematical Programming Model and Enhanced
Simulated Annealing Algorithm for the School
Timetabling Problem
O. A. Odeniyi1*, E. O. Omidiora2*, S. O. Olabiyisi3* and C. A. Oyeleye4*
1
Department of Computer Science, Osun State College of Technology, Esa-Oke, Nigeria.
2
Department of Computer Engineering, Ladoke Akintola University of Technology, Ogbomoso,
Nigeria.
3
Department of Computer Science, Ladoke Akintola University of Technology, Ogbomoso, Nigeria.
4
Department of Information Systems, Ladoke Akintola University of Technology, Ogbomoso, Nigeria.
Authors’ contributions
This work was carried out in collaboration among all authors. Author OAO formulated the
mathematical programming model and Enhanced Simulated Annealing (ESA) algorithm for solving the
school timetabling problem and wrote the first draft of the manuscript. Authors EOO and SOO
implemented and validated the formulated mathematical model and ESA algorithm using Matrix
Laboratory 8.6 software and a Nigerian high school dataset respectively. Author CAO managed the
literature searches, references and does the final manuscript. All authors read and approved the final
manuscript.
Article Information
DOI: 10.9734/AJRCOS/2020/v5i330136
Editor(s):
(1) Dr. G. Sudheer, GVP College of Engineering for Women, India.
Reviewers:
(1) Pasupuleti Venkata Siva Kumar, Vallurupalli Nageswara Rao Vignana Jyothi Institute of Engineering & Technology, India.
(2) Tajini Reda, École Nationale Supérieure des Mines de Rabat, Morocco.
(3) Marco Benvenga, UNIP Universidade Paulista, Brazil.
(4) David Lizcano, Madrid Open University, Spain.
Complete Peer review History: http://www.sdiarticle4.com/review-history/55744
Received 20 February 2020
Original Research Article Accepted 27 April 2020
Published 02 May 2020
ABSTRACT
Despite significant research efforts for School Timetabling Problem (STP) and other timetabling
problems, an effective solution approach (model and algorithm) which provides boundless use and
high quality solution has not been developed. Hence, this paper presents a novel solution approach
for solving school timetabling problem which characterizes the problem-setting in the timetabling
problem of the high school system in Nigeria. We developed a mixed integer linear programming
model and meta-heuristic method - Enhanced Simulated Annealing (ESA) algorithm. Our method
_____________________________________________________________________________________________________
*Corresponding authors: E-mail: olufemiodeniyi@gmail.com, eoomidiora@lautech.edu.ng, soolabiyisi@lautech.edu.ng,
caoyeleye@lautech.edu.ng;Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
incorporates specific features of Simulated Annealing (SA) and Genetic Algorithms (GA) in order to
solve the school timetabling problem. Both our solution approach and SA approach were
implemented using Matrix Laboratory 8.6 software. In order to validate and demonstrate the
performance of the developed solution approach, it was tested with the highly constrained school
timetabling datasets provided by a Nigerian high school using constraints violation, simulation time
and solution cost as evaluation metrics. Our developed solution approach is able to find optimal
solution as it satisfied all the specified hard and soft constraints with average simulation time of
37.91 and 42.16 seconds and solution cost of 17.03 and 18.99, respectively, for JSS and SSS to
the problem instance. A comparison with results obtained with SA approach shows that the
developed solution approach produced optimal solution in smaller simulation time and solution
cost, and has a great potential to solve school timetabling problems with satisfactory results. The
developed ESA algorithm can be used for solving other related optimization problems.
Keywords: School timetabling problem; model; mixed integer linear programming; meta-heuristics;
genetic algorithm; simulated annealing.
1. INTRODUCTION complexities (a large search space of the
problem – large number of events required to be
The School Timetabling Problem usually denoted assigned to resources while satisfying a large list
as High School Timetabling Problems (HSTP) is of constraints; increasing number of courses/
concerned with the process of fixing a sequence subjects that are very diverse for the different
of meetings between teachers and classes (set countries to be scheduled; a varying structure in
of lessons) to a specific number of timeslots different high schools even in the same country
within a prefixed time period (typically a week), or educational systems and a wider variety of
satisfying a set of constraints of various kinds many different requirements (both objectives and
[1,2,3]. These constraints are usually classified constraints) which are usually institution-specific
into two types, hard and soft. Hard constraints that changes day by day) [5,15,16,17,18].
must be satisfied in order to provide a feasible Furthermore, [19] confirmed that every model
solution, whereas, soft constraints which express has limited use and many problems are still not
the preferences and the quality of the timetable solved efficiently or optimally as most of these
can be violated (but must be satisfied as far as solution approaches however generated feasible
possible) [4]. The quality of a timetable is or low quality solutions. Even [20] opined that
measured based on how well the soft constraints finding a feasible solution to HSTP is often
have been satisfied. The more soft constraints difficult especially when the resources are tight
are satisfied, the better is the quality of the and usually results to low computational
solution considered. Further discussion on efficiency.
school timetabling can be found in [2,5].
HSTP has been intensively investigated since
Literature on school timetabling is very extensive the1960s [21,22]. Recent years have seen an
with several solution approaches (models and increased level of research activity in this area.
algorithms) proposed. In fact, many models This is evidenced (among other things) by survey
including XHSTT format (which is based on the studies, for example, [23,24,25], and the
Extensible Markup Language standard) and emergence of a series of international
several NP-complete variants of HSTP have conferences on the Practice and Theory on
been proposed in the literature, which differ due Automated Timetabling (PATAT), the Multi-
to the difference in educational system of each disciplinary International Scheduling Conference:
country, context of the application, the school Theory and Application (MISTA) and the
and the place where it is located. Representative establishment of a European Association of
literature include [6,7,8,9,10,11,12,13,14]. In Operational Research Societies (EURO) working
addition, a variety of heuristic (analytical) and group on automated timetabling [26]. However,
complete methods have been proposed to solve the international conference series on PATAT
HSTP including hybrid or enhanced which is has contributed largely to the area, for example
surveyed by [3]. [10,18,27,28,29].
Nevertheless, developing good models and Graphs, Networks and Integer Programming (IP)
algorithms for solving HSTP is a challenging task have proven to be useful in the mathematical
due to its inherent problem characteristics and formulation and solution of school timetabling
22Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
models [15,30]. Integer Programming (IP) has while Simulated Annealing (SA) and Iterated
been used to model problems with more Local Search (ILS) were used to perform local
sophisticated requirements including educational search around the initial solution. The
timetabling problems since the very early days of experimental results produced several best
Operations Research, for example, [22,31]. known solutions for the test set. However, it was
However, IP is mainly used for timetabling due to suggested that the quality of the solution can be
its modeling strength, and not as an actual improved by the development of an augmented
solution method since only small instances, or set of (larger) neighbourhoods and a proper
simplified variants of the problem, were shown to experimental study to fine tune parameter
be solved in reasonable time. In recent times selection.
several improvements have been made in IP
solvers [32] and several IP based techniques Sorensen et al. [36] established a new hybrid
have been introduced for similar HSTPs which approach called matheuristic, combining integer
were able to provide bounds and good solutions programming and meta-heuristics to solve the
after longer runs. [7,8] surveyed some of these high school timetabling problem. The
recent contributions. experimental results illustrated that the
developed matheuristic provided promising
Over the last few years, meta-heuristics including results and the potential of hybridising
its hybrids have proven to be very effective in the mathematical programming and meta-heuristics.
optimisation literature, and particularly in school However, it was suggested that the
timetabling. Souza et al. [33] applied a hybrid computational efficiency of the developed
approach of GRASP-Tabu search algorithm to matheuristic can be improved by investigating
solve school timetabling problem in order to more advanced approaches for adjusting the size
improve the timetable’s compactness and of the neighbourhoods and selecting variables in
speeds up the process of obtaining better quality each neighbourhood.
solutions. In this work, while the partially greedy
constructive procedure was used to generate Raghavjee [37] presented genetic algorithms to
good initial solutions and diversifies the search, solve STP in order to test the effectiveness of a
the Tabu Search procedure was used to improve genetic algorithm approach in solving more than
(refine) the constructed solution (search). one type of STP and as well to evaluate the
Though quality solution was generated, it was performance of a genetic algorithm that uses an
suggested that quality of the solution can be indirect representation (IGA) with that of a
improved by considering other requirements and genetic algorithm that uses a direct
adding component which measures the representation (DGA) when solving the STP.
difference between the current and the desired Both the DGA and IGA when tested on five STPs
solution to the objective function. were found to produce timetables that were
competitive and in some cases superior to that of
Soza et al. [34] addressed the solution of other methods. However, the IGA outperformed
timetabling problems using cultural algorithms. the DGA for all of the tested STPs. The solution
The authors proposed the use of domain did not address the problem of optimality and
knowledge, both a priori and extracted during the both software and computational complexities
search to improve the performance of an which can be considered for future research. In
evolutionary algorithm when solving timetabling addition, it was recommended that future
problems. The experimental results provided research may be geared towards implementing
very encouraging results. However, the optimality more advanced techniques (more efficient
of the solution was in doubt as soft constraints algorithms) to solve the STP so as to provide
violations were not addressed. It was suggested more bases for comparison.
that the performance of the developed algorithm
can be improved by analysing the mechanisms Kristiansen [38] developed different techniques
of the simulated annealing method for to solve Multiple Timetabling Problems (High
incorporation into an evolutionary algorithm or a School Timetabling, Student Sectioning and
cultural algorithm. Meeting Planning Problem) at the Danish High
Schools. Techniques developed included
Santos et al. [35] presented a hybrid SA-ILS Adaptive Large Neighbourhood Search (ALNS),
approach to solve the high school timetabling Mixed Integer Programming (MIP) Dantzig-Wolfe
problem. The Kingston’s High School Engine decomposition in a Branch-and-Bound and
(KHE) was used to generate an initial solution Column generation with a Branch-and-Price
23Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
(B&P). The experimental results proved that all Raghavjee et al. [40] presented a genetic
the developed techniques and algorithms were algorithm selection perturbative hyper-heuristic
efficient. However, it was suggested that future (GASPHH) to solve the STP. A 2-phased
research may be geared towards investigating approach was taken, with the first phase focusing
more techniques such as hyper-heuristics and on hard constraints, and the second on soft
matheuristics to measure and improve the quality constraints. GASPHH used tournament selection
of solution and computational efficiency of the to choose parents, to which the mutation and
developed exact methods. crossover operators were applied. GASPHH was
applied to five different school timetabling
Odeniyi et al. [2] developed a modified simulated problems. The performance of GASPHH was
annealing (MSA) algorithm to solve STP. The evaluated and compared to that of other methods
implementation of the MSA was characterized by applied to these problems, including a GA that
cooling schedule modification with the was applied directly to the solution space.
introduction of a non-linear factor to the GASPHH produced feasible timetables for all
exponential cooling schedule of classical problem instances, provided a generalized
simulated annealing (α = (1/log(1+t)) to become solution to the STP, and performed better than
2
parabolic (as α = (1/log(1+t+t )), in order to other methods applied to the same set of
decrease the sensitivity to the initial values problems. However, GASPHH was computa-
(parameters) that accounted for the excessive tional intensive in comparison to SGA based
convergence time of classical simulated benchmark heuristics. It was suggested that
annealing. MSA when tested on some variants of future research may be geared towards
the STP that originated from Nigerian secondary investigating approaches to reduce both the
educational institutions produced conflict-free simulation time and computational complexity as
school timetables which satisfied all hard well to improve the computational efficiency of
constraints and optimally minimised all soft the developed GASPHH.
constraints violations in less simulation time and
less solution cost. However, the solution did not Demirovic [6] presented SAT-based approaches
address the problem of software and for HSTP. The problem of determining whether a
computational complexities which can be propositional logic formula has a solution is
considered for future research. In addition, it was called the satisfiability problem (abbreviated as
recommended that future research may be SAT). The thesis explored the relation between
geared towards improving the quality of the propositional logic and HSTP, as well as related
solution of the developed algorithm by approaches. The thesis described the general
implementing more advanced techniques such HSTP (XHSTT), introduced a formal definition,
as efficient hybridised algorithms. as well two different modelling approaches, SAT-
and bitvector based. The author combined local
Dorneles [39] developed techniques that search and large neighbourhood search to solve
combine mathematical programming and XHSTT instances which were modelled as
heuristics, so-called matheuristics (fix-and- maxSAT. Each of the described methods is
optimize heuristic combined with a variable vastly different from each other and represented
neighbourhood descent strategy and column distinct ways to tackle XHSTT. The
generation approach), to solve efficiently and in a computational results showed that the developed
robust way some variants of the High School models and solution methods were competitive
Timetabling Problem (HSTP) that originated from with the state-of-the-art. However, it was
Brazilian institutions. The performance of the suggested that future research may be geared
developed matheuristics were evaluated and towards investigating techniques to find and
compared with the state-of-the-art MIP solvers prove optimal solutions in reasonable time for
designed for solving the GHSTP, referred as XHSTT in many instances.
GOAL and SVNS. The experimental results
provided very encouraging results which A review of the above related works with
indicated that the state-of-the-art MIP solvers literature analysis revealed most of these
were not efficient for solving instances of the solution approaches however generated feasible
+
HSTP . It was suggested that future research or near-optimal solutions with low computational
may be geared towards reducing the simulation efficiency. Therefore, this research focuses on
time while improving both the quality of the developing a more efficient solution approach,
solution and computational efficiency of the utilising the strength and minimising the
developed matheuristics. weaknesses of two well-known meta-heuristics,
24Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
Simulated Annealing (SA) and Genetic Algorithm The aim of this work is to develop an effective
(GA). SA is based on an analogy to solution approach (model and algorithm) which
thermodynamics simulating the cooling of a set provides boundless use and high quality solution
of heated atoms. This technique starts its search to the school timetabling problem. The overall
from any initial solution which can be generated idea is to overcome the problem of low solution
randomly or by using solutions generated by quality (slow convergence speed), which is a
another algorithm [2,41]. The main procedure common problem when using simulated
consists of a loop that randomly generates, at annealing (SA). The developed solution
each iteration, one neighbour s′ of the current approach was implemented using Matrix
solution s. Movements are probabilistically Laboratory 8.6 software. In order to validate
selected considering a temperature T and the and demonstrate the performance of the
cost variation of the movement ∆ [42]. proposed solution approach, it was tested
with the highly constrained school timetabling
SA is known for its power to avoid local optima datasets provided by a Nigerian high school,
and its theoretical guaranty to find the global using computation time, solution cost and
optimum solution when the initial temperature is constraints violation as evaluation metrics.
high enough and cooling rate is infinitely slow
[43,44,45] but converges at excessive time Developing models and algorithms that
especially when the search space is large [46]. automatically generate high quality timetables is
Several attempts have been made to speed up of great importance [6] and is still an active field
this process, such as treatment or modification of of research as well as important issues in this
the temperature parameter, which is known as domain. High quality timetables are relevant for
annealing schedule or strategy [2], and hybridiza- financial and pedagogical reasons. High quality
tion with other techniques [47,48,49,50], among timetables directly influence the quality of
others. teaching and learning, working conditions of
teachers as well as the maintenance of the
GA is a biologically motivated adaptive meta- satisfaction of students and staff, among other
heuristic that is based on natural selection and things, which leads to an overall better use of
genetic recombination. It utilizes selection, resources and learning environment [12,53].
crossover and mutation mechanisms to evolve Conversely, timetables construction by hand is
the population. The usage of GA in obtaining the usually very difficult, time consuming, and error
optimal parameters for a kernel function, its prone. Consequently, assisting the high school
flexibility, coupled with its demonstrated planners with efficient algorithms and decision
capability of searching or exploring large search support software that will automatically generate
spaces [51,52], its demonstrated ability to reach high quality timetables and/or spend less time is
near-optimum solutions to large problems, and of great importance [6,7]
its power to discover good solutions rapidly for
difficult high-dimensional problems makes the Furthermore, the high schools administration
technique an ideal candidate for consideration in requires a model of the HSTP which is
solving the timetabling problem. general enough to suit many different
requirements, and which is also tractable by
In this work, we extend previous studies about computer aided solution methods. This
the HSTP. We propose a mathematical supports the recent trend of developing general
programming model and meta-heuristic method models for HSTP, for example [6,9,10,12,
to construct school timetables. We utilise mixed 14,16,17,54,55,56,57,58]. In addition, the
integer linear programming formulation to model mathematical programming model formulated in
the problem to be solved but due to the limitation this paper is the first for high school system in
of a mathematical programming approach to Nigeria, and as well as among the most
solve large problem instances and slow comprehensive models of school timetabling to
convergence speed of classical simulated be found in the literature. Although the developed
annealing (SA) algorithm, we propose a new model is tailored to the Nigeria case but can
meta-heuristic method, called Enhanced easily be adapted to other variants from other
Simulated Annealing (ESA) algorithm which countries and other related optimization
incorporates best features of Simulated problems. The model is 'complete' in the sense
Annealing (SA) and Genetic Algorithm (GA) to that it contains all relevant practical constraints
find quality solutions to some variants of the available in the context of high school timetabling
HSTP. problem in Nigeria.
25Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
2. METHODOLOGY In addition, each event defines how lessons are
distributed over a week by requesting an amount
In this work, mixed integer linear programming of double lessons, restricting the daily limit of
technique was used to formulate the lessons, and defining whether lessons taught on
mathematical programming model of the school the same day are consecutive or not.
timetabling problem for the high school system in
Nigeria (HNSTP). An Enhanced Simulated A feasible timetable has a timeslot assigned to
Annealing (ESA) algorithm for solving the school each lesson of events satisfying
timetabling problem was formulated, which the hard constraint requirements HC1-HC13
incorporates specific features of Simulated below:
Annealing (SA) and Genetic Algorithms (GA).
The formulated mathematical model with ESA HC1: The number of lessons that each teacher
algorithm and SA algorithm were implemented must give to each class must be met
using Matrix Laboratory 8.6 software on an (Lesson Requirement Constraint).
Intel(R) Core(TM) i3-7100 CPU with 2.50GHz
speed, 32GB Hard Disk, 4GB Random Access HC2: No two classes must be scheduled to the
Memory and 32-bit Operating System, with the same teacher at the same period (Class
Window 7 operating system, and validated using Clashes Constraint).
a highly constrained Nigerian high school
dataset. The implemented algorithms were HC3: No two teachers must be scheduled to
evaluated using constraints violation, simulation the same class at the same period
time and solution cost as performance metrics. (Teacher Clashes Constraint).
HC4: A lesson cannot be scheduled to periods
2.1 Problem Specification and Model
where the teacher is unavailable
Formulation (Teachers’ Unavailability Constraint).
The goal of the High School Timetabling Problem HC5: A lesson cannot be scheduled to periods
in high school system in Nigeria (NHSTP) is to where the class is unavailable (Classes’
build a weekly timetable. The week is organized Unavailability constraint).
as a set of days D, and each day is split into a
set of periods P. All Periods are distributed in D HC6: Each class, for a given set of periods,
week days and H daily periods which occur must be involved in one lesson (One
during the same shift, that is, P = D ∗ H. Let C be Class One Lesson Constraint).
a set of classes (class-sections), T a set of
teachers and S a set of subjects. A class k ∈ C is HC7: Every teacher may be assigned at most
a disjoint group of students that follow the same one subject and one class (class-
subjects, and no idle time periods during the section) in a given period with the
week, and each subject of a class is taught by exception of indicated subjects that
only one teacher who is previously determined. A require more than one instructor
timeslot is a pair, composed of a day and a class (Uniqueness Constraint).
period (i,j), with i ∈ D and j ∈ P wherein all
periods have the same duration. Teachers l ∈ T HC8: All subjects in the curriculum of a class-
may be unavailable in some timeslots. section should appear in the timetable
for the required number of teaching
The inputs for the NHSTP are a set of events (or periods (Completeness for Students
meetings) E that must be scheduled, C*T matrix Constraints).
of lesson requirements R = (Rlk) where Rlk is the
number of periods teacher l is required to meet HC9: All subjects assigned to a given teacher
class k for the duration of the timetable should appear in the timetable for the
(containing a total of P periods in a set P), and required number of teaching periods
teachers and classes unavailability’s binary (Completeness for Teachers constraint).
matrices Tij and Ckj. Each event requires a class
k(e) ∈ C, a teacher l(e) ∈ T, and a number of HC10: The teaching periods assigned to a given
weekly hours hours(e) ∈ N. Particularly in the subject over a whole week should add
Nigerian context, a teacher, a class, and a room up to the weekly requirements for the
are pre-assigned to each event e such that specific subject (Completeness for
classrooms are not considered in the scheduling. Subjects Constraint).
26Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
HC11: Some pairs of lessons must be period per day of the week, unless it
scheduled simultaneously (Simultaneity requires more teaching periods than the
Constraint). days of the week or there is a special
request for multiple or consecutive
HC12: Certain subjects to be taught in multi- hours, in which cases they are
period slots at most once a day for a scheduled accordingly (Uniform
given class section must be followed distribution of subjects Constraint).
(Consecutiveness constraint).
SC8. Pre-assignments of certain subjects to
HC13: The timetable of every class-section specific time periods must be respected
should not carry empty slots during the (Pre assignments of certain subjects
week (Compact Student Schedules Constraint).
Constraint).
Besides feasibility regarding hard constraints, The notation used in the problem specification
as many as possible of the soft and the formulated model for NHSTP considering
requirements SC1-SC8 stated below should be all the hard and soft constraints requirements
satisfied: listed above are presented as follows.
SC1: More lessons to the same class in the Sets
same day than the maximum specified
for the pair teacher-class must be i∈D days of week. D = {1,2,...,|D|}.
avoided (Classes’ maximum workload
per day constraint). j∈P periods of a day. P = {1,...,|P |}.
SC2: The specified maximum number of daily P1 P without the last two periods of a day.
lessons of each teacher must be P1 = {1,...,|P | −2}.
respected (Teachers’ maximum
workload per day constraint). k∈C set of classes.
SC3: Assignment of teachers to periods in l∈T set of teachers.
which teachers would prefer not to teach
must be avoided (Teachers’ Preference m ∈ S set of subjects.
or undesired Constraint).
e∈ E set of events.
SC4: Avoid teachers’ idle periods. As much as B set of quadrupes of the form (l1; k1; l2; k2)
possible, minimise the number of empty with l1 ≠ l2 and k1 ≠ k2 such that all
or holes periods between two lessons of teacher l1 to class k1 must be
consecutive periods where a teacher is simultaneous to lessons of teacher l2 to
assigned to a class. Breaks and free- class k2
time periods are not considered as idle
periods or holes (Teachers’ Idle Period Skl = {mS: m is a subject that teacher l
Constraint). teaches for class-section k}
SC5: Teachers’ request for double lessons
S*kl = {mS: m is any regular subject that teacher
(lessons conducted in two consecutive
l teaches for class-section k}, where the term
periods) should be granted as often as
“regular” refers to all subjects that do not require
possible (Double lesson per week
any special type of scheduling.
constraint).
SC6: Teachers’ request for parallelism of SlSim = {(m, k, l*): mS is a subject that teacher
subjects (subjects scheduled for the lT teaches for a part of section k
same time periods) should be granted as simultaneously with another subject taught by
often as possible (Parallelism of subjects the “basic” teacher l*}
Constraint).
SlCol = {(m, k, l*): mS is a subject that teacher
SC7: Lessons for any given subject must be lT teaches for section k in collaboration with
scheduled for at most one teaching the “basic” teacher l*}
27Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
Scons = {(m, k, hm): mS is a subject of section l,i
k that needs to be scheduled in block(s) of hm : The number of idle times at the agenda of
consecutive periods} teacher l on day i
Sparal = {[(ma, mb); (ka, kb); (la, lb)]: maS is a TTP: Total Time Periods. TTP indicates the total
subject that teacher la teaches for section ka that number of time periods to be scheduled in the
needs to be scheduled always in parallel to mb a timetable.
subject taught by teacher lb for section kb}
WTLl: Weekly Teaching Load for teacher l. WTLl
Sexcl = {[(ma, ka, la; mb, kb, lb; ...;mw, kw, lw)
indicates the total number of time periods to be
(la)]:ma, mb, ..., mwS are subjects that teachers assigned to teacher l each week.
la,lb,..., lw T teach to sections ka, kb, ..., kw,
respectively, however no more than a certain DTLl: Average Daily Teaching Load for teacher l.
number of them may be scheduled in the same DTLl indicates the daily average number of time
day or time period} periods assigned to teacher l.
fix
S = {(ia, ja, ma, ka, la): maS is a subject that WTLkl: Weekly Teaching Load of teacher l for
teacher la teaches to section ka and should be class-section k. WTLkl indicates the total number
scheduled on day ia and in period ja} of time periods to be assigned to teacher l for
class-section k each week.
Dl = { iD : i is any day of the week for which
teacher l is available for the school} WTLklm: Weekly Teaching Load of teacher l for
class-section k and subject m. WTLklm indicates
Cl = { kC : k is a class-section of the school to the total number of time periods to be assigned
which teacher l teaches at least one subject} to teacher l for subject m each week.
Parameters TTPk: Total Time Periods for section k. TTPk
indicates the total number of time periods over all
Rlk: The workload of an event (l,k), that is, the subjects of class-section k to be assigned each
number of lessons that must be taught by the week.
teacher l for the class k.
Hkmax: Is a parameter that indicates the maximum
λ: The maximum number of permitted lessons number of teaching periods that section k may
per day have during any day of the week. Hkmax may be
set equal to J, the length of each day, however,
m : Is the number of the multi-period slots
in order to create more balanced timetables for
the classes, it is preferable to set a different
(double lessons) required for subject m every upper limit for each section of the school.
week. Therefore, Hkmax equals to [TTPk/D]. In general,
however, it holds that Hkmax P, kC.
l ,i
: The maximum number of double lessons Variables
requested by teacher l with class k.
x i, j, k, l, m: Binary variable that indicates whether
l,i subject m, taught by teacher l to the class section
th
: The effective number of allocated double k, is scheduled for the j period of day i.
lessons.
k,l,i xi, j ,k ,l ,m : Complement of the binary variable x i,
: The total number of lessons allocated for
class k with teacher l on day i. xi , j ,k ,l ,m =
j, k, l, m,
that is, x i, j, k, l, m = 0 if 1 and
vice versa.
k ,l
: The maximum number of permitted
lessons per day.
yi,tm ,k,hm ,m
: Binary variable that indicates
whether subject m, is scheduled for hm
l , j ,i consecutive periods on day i for class-section k,
: The total number of lessons allocated for st
teacher l on period j of day i. with tm being the 1 period for this assignment.
28Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
Tij: binary variable that indicates whether X = An arbitrary matrix (x i, j, k, l, m) is called a
teacher l is available to teach class k subject m timetable. X ∈ {0, 1}
scheduled for jth period of day i.
The objective function is to minimise the
Ckj : binary variable that indicates whether constraints (soft) violation that is formulated as
class k is available to be taught by teacher l solution cost function Cf(s), which associates a
th
subject m scheduled for j period of day i. cost value to a given solution. Such value was
used to compare the goodness of different
Zkj : binary matrix that indicates whether solutions. This function was defined as follows:
class k must be taught a subject m by teacher l Let S be the search space; sS, a solution; n,
th
at j period of day i, the number of type of problem constraints
considered, wi, the penalty weighting associated
with each constraint type i and vi(s), represents
ll ,i , j the number of constraint violations of type i in a
: binary variable that indicates whether
teacher l has been scheduled to teach at an solution s, vi(s) = 0 if constraint type i is satisfied
undesired period j on day i. and vi(s) = 1 if constraint type i is violated.
Therefore, the solution cost function Cf(s) is given as:
n
. Cf(s) = w v (s)
i 1
i i (1a)
n
Minimise w v (s)
i 1
i i (1b)
Subject to
P
∑ xi,j,k,l,m = Rlk i D , j P l T , k C (2)
j=1
C
∑ xi,j,k,l,m ≤ 1 i D , j P l T , m S (3)
k=1
T
∑ xi,j,k,l,m ≤ 1 i D , j P k C , m S (4)
l=1
C
∑ xi,j,k,l,m ≤ Tij i D , j P l T , m S (5)
k=1
T
∑ xi,j,k,l,m ≤ Ckj i D , j P k C , m S (6)
l=1
T
∑ xi,j,k,l,m ≥ Zkj i D , j P k C , m S (7)
l=1
x i , j , k ,l , m sim x i , j , k , l * , m 1, l T , j P , i D l
k C l m S kl* m , k , l S
*
l S lcol
(8)
x i , j , k ,l , m x i , j , k ,l * , m TTP k , k C
l T k m S kl i D l j P
l T k m , k , l * S lsim S lcol i D l j P
(9)
29Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
x i , j , k ,l , m sim x i , j , k ,l * , m WLT kl , k C , l T
i D j P m S kl* m , k ,l S *
l S lcol i D l j P
(10)
xi , j , k , l , m WTL klm , kC , lTk , mS kl
iD l jP
(11)
xl1k 1 j xl 2 k 2 j x l1k 1 j x l 2 k 2 j = 1 ([l ; k ; l ; k ] B; jP)
+ 1 1 2 2 (12)
tm hm 1
*
hm yi ,tm ,k ,hm ,m xi , j ,k ,l ,m yi ,tm ,k ,hm ,m hm 1, m, k , hm Scon,iDl , t m P hm 1
j tm
(13a)
p h m 1
y i ,t m , k , hm , m 1, m , k , h m S con , i D l
t m 1
(13b)
p hm 1
y i , t m , k , h m , m m , m , k , hm S con
i D l t m 1
(13c)
xi , j , k , l , m
l T k m S kl* *
sim
m , k , l S S lcol
xi , j , k , l * , m 1 k C , i D , j 1,...., H kmax 1
l (14a)
x i , H kmax , k , l , m sim x i , H kmax , k , l * , m 1, i D , k C
l T k m S kl* m , k , l S *
l S lcol
(14b)
x i , j , k ,l , m sim x i , j , k ,l * , m H kmax , i D , k C
l Tk m S kl* j P m ,k ,l S
*
l S lcol j P
(14c)
xi , j ,k ,l ,m k ,l ,ikC , lT , iD
kCl lTl iD
≤ (15)
xi , j , k , l , m l , j ,i lT , jP, iD
lTl iDl jP
≤ (16)
xi , j , k , l , m ll ,i , j lT , iDk , jP
lTl iDl jP
≤ (17)
l ,i
l T i D ≤ 1 l T , i D (18)
l ,i l ,i , lT , iD , l ,i l ,i
≤ l T i D (19)
xi , j ,k1 ,l1,m1 x, j ,k2 ,l2 ,m2 0, m1 , m2 ; k1, k2 ; l1, l2 S paral , iD, iP
(20)
xi , j ,k ,l ,m 1, iD , kC , lTk , mS kl
j P
(21)
xi , j ,k ,l ,m 1, i, j, k, l, mS fix
(22)
xi,j,k,l,m = 0 or 1 i D , j P , k C l T , m S (23)
30Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
Constraint set (2) ensures that the number of lessons of each teacher is respected. Constraint
lessons that each teacher must give to each set (17) ensures the assignment of teachers to
class is fully scheduled. Constraint set (3) periods in which teachers would prefer not to
ensures that no two classes are scheduled to the teach is avoided. Constraint set (18) determines
same teacher at the same period. Constraint set the number of teachers’ idle periods in a solution.
(4) ensures that no two teachers are scheduled Constraint set (19) ensures that teachers’
to the same class at the same period. Constraint request for double lessons is granted. Constraint
set (5) ensures that a lesson cannot be set (20) ensures that teachers’ request for
scheduled to periods where the teacher is parallelism of subjects is granted. Constraint set
unavailable. Constraint set (6) ensures that a (21) ensures that uniform distribution of subjects
lesson cannot be scheduled to periods where the is met. Constraint set (22) ensures pre-
class is unavailable. Constraint set (7) ensures assignments of certain subjects to specific time
that each class, for a given set of periods, are periods are respected while Constraint set (23) is
scheduled to only one lesson at a time. required to ensure the integrality of the solution.
Constraint set (8) ensures that every teacher is
assigned at most one subject and one class In this work each hard constraint was assigned a
(class-section) in a given period with the weight of 20 to stipulate their higher priority than
exception of indicated subjects that require more the soft constraints and to allow the proposed
than one instructor. Constraint set (9) ensures solution leads the search process towards valid
that all subjects in the curriculum of a class- solutions in accordance to the literature [59]. The
section appeared in the timetable for the required weight assigned to each of the soft constraints
number of teaching periods. Constraint set (10) (preferences) varies to indicate the relative
ensures that all subjects assigned to a given importance of each preferences compared to
teacher appeared in the timetable for the others such that a weight of 6 was assigned to
required number of teaching periods. SC1, SC2, SC5 and SC6; a weight of 4 was
assigned to SC7 and SC8; a weight of 3 was
Constraint set (11) ensures that the teaching assigned to SC3 and finally a weight of 1 was
periods assigned to a given subject over a whole assigned to SC4.
week added up to the weekly requirements for
the specific subject. Constraint set (12) ensures These weight set was informed by the literature
that some pairs of lessons are scheduled which stated that weighted penalty based
simultaneously. Constraint set (13) ensures that evaluation function should be used for
certain subjects to be taught in multi-period slots timetabling problems where an abundance of
at most once a day for a given class section are different constraint combinations is encountered
followed. Constraint (13a) forces hm basic [60,61,62] and thus allows for some constraints
variables xi,j,k,l,m that refer to consecutive time to have a higher priority than others. It must be
periods to take the value of 1, while constraint noted that (i) the lower the value of Cf(s) for a
(13b) ensures that only one block of consecutive given solution s, the more the quality of s, (ii)
periods may be assigned in any given day and both the distance to feasibility and the goodness
constraint (13c) indicates that there should be of the solution was measured.
exactly m of these blocks for the whole week. 2.2 Formulation of Enhanced Simulated
Constraint set (14) ensures that the timetable of Annealing (ESA) Algorithm
every class-section did not carry empty slots
during the week. Basically Constraint set (14a) The simulated annealing (SA) algorithm was
and (14b) ensure that for each class-section enhanced to form Enhanced Simulated
there is exactly one subject scheduled for any Annealing (ESA) Algorithm through the following
given period (except may be the last one) of any three stages:
day, while Constraint (14c) on the other hand,
checks whether all subjects of class-section k are (1) Modification of Simulated Annealing (SA)
scheduled within the maximum stretch allowed in terms of the temperature reduction
for the class-section. parameter in order to improve its
efficiency in terms of convergence speed
Constraint set (15) ensures that the limit set on and solution cost, by introducing a
the maximum number of lessons a class may parabolic reduction parameter (α =
have per day is met. Constraint set (16) ensures 2
(1/log(1+t+t )) as suggested by [2]. This is
that the specified maximum number of daily due to the fact that the efficiency of SA
31Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
often depends on cooling schedule and by Step 4.3: Compute the fitness E(Cj) of new
carefully controlling the rate of cooling the individual Cj, j = 1, …, n.
temperature; SA can find the global
optimum exponential faster [63,64]. Step 4.4: For j = 1, …, n, compute ∆E = E(Cj)-
(2) Integration with Genetic Algorithm (GA) in E(Pj), where E(Pj) is the fitness of parent
order to: (i) further improve the individual Pj. If ∆E ≥ 0, then produce a number r
performance of SA in terms of speed and with uniform distribution in interval [0,1]. If exp
quality of solution as suggested by [65]; (ii) (-∆E/T) > r, then replace individual Pj by child
find a good balance between the individual Cj. If ∆E < 0, then discard the child
exploitation of found-so-far elements and individual Cj.
the exploration of the search space in
order to find global optimal solution as Step 4.5: If the termination condition is satisfied,
suggested by [66] and (iii) improve its local then the whole procedure is stopped, else the
search ability as suggested by [67]. By this value of temperature T is decreased.
enhancement (integration), the genetic
operators of GA were applied to observe The algorithmic structure of SA algorithm that
the behaviour of Simulated Annealing was coded using the MATLAB Laboratory 8.6
which resulted into less parameter to software is as presented as follows:
control. In tune with the principle of the
integration, new individuals were produced Step 1: Generate an initial schedule S.
with Genetic Algorithm (GA) after which
these individuals were processed with SA Step 2: Set the initial best schedule S* = S.
while the corresponding results were
further used as the new individuals of the Step 3: Compute cost of S: C(S).
next generation.
(3) Reordering the sequence of evolutionary Step 4: Compute initial temperature T0.
operations to become (mutation, selection
and crossover) instead of (selection, Step 5: Set the Initial Temperature T = T0, set
crossover and mutation) in order to further Parabolic Temperature reduction factor
reduce the convergence time and as well as α = (1/log(1+t)), and Final
as to generally improve its computational Temperature as Tt+1 = αTt
efficiency as reported by [68,69].
Step 6: While stop criterion is not satisfied do.
The above sequence of processes resulted into
Enhanced Simulated Annealing (ESA) algorithm. (a) Repeat Markov Chain Length (M) times:
The algorithmic structure of ESA algorithm that i. Select a random neighbor S to the
1
was coded using the MATLAB Laboratory 8.6 1
current schedule, (S NS).
software is as presented as follows: ii. Set ∆ (C) = C(S1) – C(S).
Step 1: Initialize the temperature parameter T, iii. If (∆ (C) ≤ 0 {downhill move}).
that is, set T = T0, in which T0 is a large positive • Set S = S1
number, set the exponential temperature • If C(S) < C(S*) then set S* = S
2
reduction factor as α = (1/log(1+t+t )), and final iv. If (∆ (C) > 0 {uphill move}).
temperature as Tt+1 = αTt • Choose a random number r uniformly
from [0, 1]
Step 2: Produce the initial population made up of • If r < e- ∆(C)/T then set S = S1
n individuals. (b) Reduce (or update) temperature T.
Step 3: Compute the fitness of each individual in Step 7: Return the Schedule S*
the initial population.
Step 4: Repeat: 3. RESULTS AND DISCUSSION
Step 4.1: Carry out evolutionary operations, that The Simulated Annealing (SA) algorithm and the
is, mutation, selection and crossover, for developed Enhanced Simulated Annealing (ESA)
individuals in the current solution population. algorithm were tested with the highly constrained
school timetabling dataset provided by a Nigerian
Step 4.2: Let Cj be the child individual produced high school. The algorithms were tested under
by a parent individual Pj, j = 1, …, n. several optimization runs. The results of
32Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
performance evaluation of both algorithms (SA (i) Constraints violation: Constraints violation is
and ESA) are as shown in Table 1. the metric that measures the feasibility and
optimality of the solution produced by an
As depicted in Table 1, both the average algorithm. An algorithm that satisfies the
simulation time and solution cost to generate a problem’s all hard constraints is said to
high quality timetable in SA algorithm is higher produce a feasible solution. The optimality
than that to generate an optimal timetable in the (goodness/quality) of an algorithm’s solution is
developed ESA algorithm. The higher simulation indicated by how much soft constraints an
time and solution cost of SA algorithm can be algorithm satisfied. As depicted in Table 1, SA
attributed to the slow convergence speed and algorithm produced high quality timetable as a
exponential cooling schedule inherent in SA result of violation of one of the soft constraints
algorithm. The modified annealing schedule while the developed ESA algorithm produced
(slow cooling schedule - parabolic), improved optimal solution as it satisfied all the specified
local search ability and the reordered constraints (hard and soft).
evolutionary operation sequence in the
developed ESA algorithm helped to improve the (ii) Simulation Time: Simulation time is the
convergence speed which in turns reduced the parameter which measures the time utilized by is
average simulation time and solution cost to an algorithm to run until the result is produced. It
generate an optimal timetable in the developed otherwise known as computation, execution or
ESA algorithm. run time. As advocated by [73], simulation time
should be considered first when dealing with the
This result confirmed the previous literatures that performance evaluation of optimization
indicated that: high quality solutions can only be algorithms for combinatorial problems. It should
obtained if SA’s parameters (cooling schedule, be a key element of any such evaluation [74].
update moves, initial solution, among others) are Indicated that one of the most important factors
well tuned and that (i) the efficiency of SA considered before choosing the winner during
algorithm depend on cooling schedule and by the second international timetabling competition
carefully controlling the rate of cooling the (ITC-2007) was the simulation time.
temperature SA can find the global optimum
exponential faster [63,64]; (ii) the performance of Table 1 showed the obtained values of the
SA in terms of speed, quality of solution (global simulation time of both SA algorithm and the
optimal solution) and local search ability can be developed ESA algorithm for JSS and SSS
improved by integrating it with GA [65,66,67]; respectively. The simulation time of the SA
and (iii) the convergence time and computational algorithm and the developed ESA algorithm are
efficiency of GA-based algorithm can be 40.90 and 37.91 seconds respectively for JSS
improved by reordering the sequence of and 45.82 and 42.16 seconds respectively for
evolutionary operations to become (mutation, SSS. This is clear evidence that the developed
selection and crossover) instead of (selection, ESA algorithm utilized less time and converges
crossover and mutation) [68,69]. faster than the SA algorithm to produce an
optimal timetable as a result of its enhanced
The following section gives the detailed result of features (slow cooling schedule - parabolic,
the performance evaluation of the two solution improved local search ability and the reordered
approaches (SA and the developed ESA): evolutionary operation sequence).
Table 1. Constraints violation, simulation time and solution cost evaluation result
Parameters SA ESA
Junior Secondary School
Number of Hard Constraint Violated 0 0
Number of Soft Constraint Violated 1 0
Average Simulation time (Seconds) 40.90 37.91
Average Solution Cost 20.88 17.03
Senior Secondary School
Number of Hard Constraint Violated 0 0
Number of Soft Constraint Violated 1 0
Average Simulation time (Seconds) 45.82 42.16
Average Solution Cost 23.43 18.99
33Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
It was observed that the developed ESA yielded 4. CONCLUSION
high convergence speed which resulted into the
lower average simulation time in producing The high school timetabling is a classical
optimal solution, but utilised more time to combinatorial optimization problem that takes a
compute the timetables for SSS classes than large number of variables and constraints into
JSS classes because of the different subject account. Due to its combinatorial nature, solving
groups that exist in SSS, such as Science group, medium and large instances to optimality is a
Commercial group and Art group.. This result challenging task. Specifically, school timetabling
confirmed the literatures that by carefully problem is perhaps the most difficult problem
controlling the rate of cooling the temperature, which high schools face in Nigeria. In this paper,
SA can find the global optimum exponential we presented a novel solution approach for
faster since slow cooling schedules are solving a highly constrained school timetabling
generally more effective [63] and that the problem which characterizes the problem-setting
performance of SA in terms of speed and quality in the timetabling problem of the high school
of solution can be improved by integrating it with system in Nigeria which has not been completely
GA [65,67]. addressed in the literature. We presented a new
mathematical programming model of timetabling
(iii) Solution cost: The quality of a timetable is for high schools in Nigeria using mixed integer
defined by a solution cost, otherwise known as linear programming formulation for which the
fitness value or solution quality function. The current timetable is considerably improved. In
fitness function calculates the number of addition, we presented a meta-heuristic method,
constraint breaches (usually with a weighted an Enhanced Simulated Annealing (ESA)
value) for different constraints. This value is used algorithm that incorporates specific features of
as measurement for quality of the timetable Simulated Annealing and Genetic Algorithms for
generated by the algorithms. It is used to solving the problem. Results show that our
compare the goodness of different solutions as approach provides high quality solutions (optimal
better timetables are produced the better fitness timetables) in smaller computational time and
values emerges. It is implicitly defined through solution cost when compared with results
the school timetabling problem specification and obtained with SA approach.
constraints as given in Equation 1a. The lower
the value of solution cost for a given solution the The results of this work confirmed previous
more the quality of the solution [70]. research reports that high quality solutions can
be obtained if SA’s parameters are well tuned as
Table 1 showed the measured values of the the analysis of the results showed that though
solution cost of the two algorithms (SA and the both the SA algorithm and developed ESA
developed ESA) for Junior Secondary School algorithm yielded better quality solutions when
(JSS) and Senior Secondary School (SSS) compares to manual allocation procedures but
respectively. The average solution cost values of the developed ESA algorithm yielded higher
SA algorithm and the developed ESA algorithm quality solutions. The observed high quality
were 20.88 and 17.03 respectively for JSS, and solutions provided by the developed ESA
23.43 and 18.99 respectively for SSS. This is algorithm resulted from its tuned parameters -
clear evidence that the developed ESA returned modified annealing schedule (slow cooling
the best optimal solution (with lower solution cost schedule - parabolic), improved local search
value), but with higher value for SSS because of ability and the reordered evolutionary operation
the somewhat more complicated structure of sequence. Furthermore, the result shows that the
student groups and the demand for compact developed solution method (ESA algorithm) is
scheduling in SSS than JSS. This is attributed to very promising to solve the school timetabling
the slow temperature reduction component problem, motivating its use to variants of this
(parabolic cooling rate of the developed ESA problem, as well as to other general
algorithm) since the solution cost is generally combinatorial optimization problems.
improved with slow cooling rate. This result
confirmed the literatures that the choice of the A possible future research area is to develop
cooling schedule influences the quality of other solution methods that might solve the
solution obtained with SA [71,72] and that that problem more efficiently. For example, the
slow cooling schedules are generally more efficiency of the developed solution method can
effective, and also that the solution cost generally be improved by incorporating it within an
improves with slower cooling rates [63]. evolutionary algorithm or a cultural algorithm..
34Odeniyi et al.; AJRCOS, 5(3): 21-38, 2020; Article no.AJRCOS.55744
Another future research area is to incorporate Logic and Search (a SAT/ICLP workshop
additional requirements that might be required by at FLoC 2014); 2014a.
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