Solving Examination Timetabling Problems through
Adaptation of Heuristic Orderings
E.K.Burke*
J.P.Newall+
*
Automated Scheduling, Optimisation and Planning Group
School of Computer Science & IT
University of Nottingham
Nottingham NG8 1BB, UK
ekb@cs.nott.ac.uk
+eventMAP
Limited
21 Stranmillis Road
Belfast BT9 5AF
Northern Ireland
Jim.Newall@eventmaponline.com
Keywords: examination timetabling, heuristics, graph colouring
AMS(MOS) classification: 68W99, 90C59
ABSTRACT
Heuristic ordering based methods, very similar to those used for graph colouring problems,
have long been applied successfully to the examination timetabling problem. Despite the
success of these methods on real life problems, even with limited computing resources, the
approach has the fundamental flaw that it is only as effective as the heuristic that is used. .
We present an adaptive method that adapts to suit a particular problem instance “on the
fly”. This method provides an alternative to existing forms of ‘backtracking’, which are
often required to cope with the possible unsuitability of a heuristic. We present a range of
experiments on benchmark problems to test and evaluate the approach. In comparison to
other published approaches to solving this problem, the adaptive method is more general,
significantly quicker and easier to implement and produces results that are at least comparable (if not better) than the current state of the art. We also demonstrate the level of generality of this approach by starting it with the inverse of a known good heuristic, a null ordering
and random orderings, showing that the adaptive method can transform a bad heuristic
ordering into a good one.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
2
1 Introduction
The exam timetabling problem is essentially concerned with scheduling a number of exams into a limited number of timeslots or periods in order to satisfy, as much as possible, a set of specified constraints. These constraints vary from institution to institution. A detailed analysis and study of
institutional requirements in over 50 British universities is presented in [Burke et al., 1996a]. It is often
essential that some constraints are completely satisfied. Such constraints are called hard constraints.
Usually these constraints relate to operational limitations that cannot be bypassed in the real world,
such as the constraint that one person cannot be in two places at once or that there is a maximum number of people that can be accommodated in a particular room. We call a timetable that satisfies all hard
constraints a feasible timetable.
Another class of constraints that occur in timetabling problems are those that are deemed desirable, but
that are often either difficult or impossible to fully satisfy. This could include providing study time for
each student between any two exams, or making more efficient use of rooms. These constraints are
usually called soft constraints. Such constraints often determine the quality of a timetable. In general
we would think of a good quality timetable as one that is (firstly) feasible and that (secondly) satisfies
the soft constraints to an acceptable level. Of course, the quality of a solution is very much subjective.
One institution’s idea of a good timetable could very well be a poor timetable for another institution.
For example, it may be that one institution insists on having a clear day in between exams for all of its
students (i.e. it makes this a hard constraint). Another institution may be more concerned with holding
all of the exams as quickly as possible in which case the inclusion of the above constraint would be detrimental to the quality of the timetable. Issues surrounding the perceived quality of examination timetables in real world situations (British Universities) are discussed in more detail in [Burke et al., 1996a].
Over the last forty years or so, there has been a wealth of literature on automated timetabling and there
have been several review papers that discuss the major approaches to timetabling [Bardadym, 1996,
Burke et al., 1997, Carter, 1986, Carter and Laporte, 1996, de Werra, 1985, Schaerf, 1999]. The early
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
3
approaches to exam timetabling tended to employ heuristic ordering where a heuristic is used to measure the difficulty of scheduling a particular exam [Foxley and Lockyer, 1968]. These heuristics are
often based upon graph colouring heuristics [Burke et al., 1994, de Werra, 1997, Mehta, 1981, Welsh
and Powell, 1967]. The overall idea is that the difficult exams are scheduled first to get them out of the
way and the easier exams are scheduled towards the end of the process. This general approach has
proved to be very effective, particularly when backtracking is added to the process [Carter et al., 1994].
The backtracking procedure is called upon when it is not possible to place a particular exam into the
timetable because of earlier placements. In essence, the procedure unschedules certain exams that clash
with the current exam. This should then allow the problem exam to be scheduled. The recently
unscheduled exams can then be rescheduled at alternative times. More details about these heuristic procedures can be found in the survey papers mentioned above. Carter, Laporte and Lee in 1996 [Carter
et al., 1996] presented some very competitive results on a range of benchmark problems. These results
are generated by the employment of graph colouring based heuristics and backtracking. Burke, Newall
and Weare [Burke et al., 1998b] did not employ backtracking but incorporated a random element into
the heuristic procedure to produce a compromise approach between heuristic ordering methods and the
more time consuming meta-heuristic approaches.
In recent years a range of these meta-heuristic approaches have been applied to examination timetabling. Simulated annealing for the problem has been investigated by Thompson and Dowsland
[Thompson and Dowsland, 1996a, Thompson and Dowsland, 1996b, Thompson and Dowsland, 1998,
Dowsland, 1996, Dowsland, 1998]. They employ a 2 phase approach. The 1st phase is concerned with
finding a feasible solution, while the 2nd phase attempts to optimise the soft constraints in an attempt
to produce better quality timetables. Bullnheimer [Bullnheimer, 1998] also investigated simulated
annealing for examination timetabling. He concentrates on small scale problems and one real world
problem in particular. Tabu search has also been extensively studied. One of the earliest investigations
of this meta-heuristic for exam timetabling was presented by Hertz in 1991 [Hertz, 1991]. Boufflet and
Nigre employed tabu search to successfully solve a real world problem at the University of Technology
of Compiegne [Boufflet and Negre, 1996]. White and Xie [White and Xie, 2001] present a tabu search
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
4
based approach to examination timetabling. Their approach uses both short term memory and longer
term memory in an attempt to find good quality solutions. Their results on two data sets compare
favourably with the results of Carter, Laporte and Lee [Carter et al., 1996] and with those reported by
Di Gaspero and Schaerf [Di Gaspero and Schaerf, 2001] who also investigate a tabu search method for
exam timetabling. Di Gaspero and Schaerf’s approach draws heavily on graph colouring based heuristic methods. They present a relatively detailed comparison of their approach with other approaches
(which are presented in [Burke and Newall, 1999, Burke et al., 1996b, Carter et al., 1996]) on a range
of benchmark problems and produced very competitive results. The authors say, “...our approach
works better than the pure memetic algorithm and the constructive one. Only the approach based on
decomposition performs better.”. The memetic and decomposition methods are briefly discussed
below.
Over the last few years there has been a significant level of interest in evolutionary and genetic
approaches to examination timetabling. In 1993, Corne, Fang and Mellish [Corne et al., 1993] investigated a straightforward genetic algorithm approach for the examination problem at the Department of
Artificial Intelligence at the University of Edinburgh. Their approach compared favourably with previously produced manual results. Corne, Ross and Fang [Corne et al., 1994] discussed the employment
of evolutionary algorithms for exam timetabling problems and noted the potential of evolutionary algorithms in this area. They also discussed and presented mutation operators and a delta-evaluation
method to speed up the evolutionary approach [Ross et al., 1994]. Burke, Elliman and Weare [Burke
et al., 1995b] presented and discussed a series of recombination operators for exam timetabling. Ross
and Corne [Ross and Corne, 1995] compared three approaches (genetic algorithms, simulated annealing and stochastic hill climbing) on a test suite of 5 exam timetabling problems. The authors point out
that this was a rather limited experiment but that simulated annealing and stochastic hill climbing work
better than the genetic algorithm in terms of solution quality. Of course, as Ross and Corne point out,
the representation used in the genetic algorithm is a particularly important consideration and that comparisons employing different representations may lead to different results. Burke, Elliman and Weare
[Burke et al., 1995a] represent the timetable directly and incorporate graph colouring techniques into
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
5
the crossover operators. Hybridisation of heuristics and meta-heuristics to solve the problem is a theme
that runs through several later papers on the exam timetabling problem. Burke, Newall and Weare
[Burke et al., 1996b] presented a memetic algorithm and applied it to benchmark problems. This
mematic approach was a hybrid evolutionary algorithm that employed mutation only (no recombination) and a hill-climbing algorithm. It produced good results (at the time) on benchmark data. Ergul
[Ergul, 1996] presented a genetic algorithm based exam timetabling approach and demonstrated its use
on real data from the Middle East Technical University in Ankara, Turkey. Corne and Ross [Corne and
Ross, 1996] looked at “Peckish” initialisation strategies. The term peckish is used to represent slightly
hungry algorithms rather than greedy ones. They showed that peckish strategies are more effective than
greedy or random ones on exam timetabling data from the Department of Artificial Intelligence at the
University of Edinburgh. In 1998, Burke, Newall and Weare [Burke et al., 1998a] presented a study of
initialisation strategies for evolutionary exam timetabling. In particular, they concentrated upon the
employment of graph colouring heuristics in the initialisation process and showed that such strategies
can improve the performance of a memetic approach. The authors pointed out that the algorithm with
initialisation often started out with solutions which were comparable to the solutions that it finished
with when started on a random population. This meant that the evolutionary algorithm could be
employed to “fine tune” the solutions.
In 1999, Burke and Newall [Burke and Newall, 1999] incorporated a problem decomposition method
with a memetic algorithm. The basic idea was to split up large timetabling problems into a series of
smaller subproblems and deal with each one in turn. The obvious drawback is that this has the potential
to run into difficulties with later sub-problems because of decisions taken in earlier subproblems. The
authors addressed this by using graph colouring heuristics to form the subproblems and by employing a
look-ahead approach as each of the sub-problems is dealt with. The overall strategy is that the “hardest” exams to schedule (according to the heuristics) are placed into the earlier sub-problems. The
development of the decomposition method was motivated by the goal of trying to speed up the evolutionary process. However, it also significantly improved the solution quality and produced the best
results that have so far been published on certain benchmark problems. A similar concept has been
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
6
employed by Carter and Johnson [Carter and Johnson, 2001] with an initialisation method based on the
identification of cliques in the underlying graph, including both non-maximal cliques and quasicliques. Erben [Erben, 2001] has further investigated the theme of incorporating graph colouring heuristics and evolutionary methods for exam timetabling problems. This approach uses a representation
that draws on the “grouping” character of graph colouring. For further details about grouping problems
see [Falkenauer, 1998]. Erben’s method was developed for the graph colouring problem and initial
modifications to apply it to the exam timetabling problem have shown promise. Burke, Bykov and
Petrovic [Burke et al., 2001] drew on the hill climbing and mutation operators from [Burke and
Newall, 1999, Burke et al., 1996b] to develop a multi-criteria approach to examination timetabling. It
is not possible to compare such approaches with methods that employ a single cost function but they
provide a higher level of flexibility in the handling of constraints. The multi-criteria approach is able to
comfortably handle a range of fundamentally different constraints and to establish a balance between
them according to whatever quality measure may be applied.
In 1997, Ross, Hart and Corne [Ross et al., 1998] discussed some of the limitations of genetic algorithms and concluded that a future direction for timetabling research might be to investigate genetic
algorithms to choose the right algorithm to solve the given problem rather than being employed
directly on the problem itself. In 1999, Terashima-Marin, Ross and Valenzuala-Rendon [TerrashimaMarin et al., 1999] investigated just such an approach and demonstrated the potential that this shows to
raise the level of generality of automated timetabling methods. Indeed this is one of the research
themes that the authors are currently investigating with Ross and Hart on a UK research council funded
project [Burke et al., 2000] to investigate hyper heuristics which can be thought of as “heuristics to
choose heuristics”. The research presented in this paper has been carried out under this project.
Another approach to exam timetabling over the years has been to consider constraint based techniques.
In 1994 Nuijten, Kunnen, Aarts and Dignum [Nuijten et al., 1994] employed constraint satisfaction on
a real world exam timetabling problem from Eindhoven University of Technology. Boizumault, Delon
and Peridy [Boizumault et al., 1996] used a constraint programming language called CHIP to develop
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
7
an exam timetabling package. They also presented results on a real world timetabling problem from
L’Universite Catholique de l’Ouest.
In 1997, David [David, 1998] presented a constraint based
approach that incorporates the use of heuristics and local repair methods. His approach was used at the
Ecole des Mines de Nantes.
Examination timetabling problems are generating considerable interest from researchers across the
fields of Operations Research and Artificial Intelligence. More details about some future directions for
timetabling research such as case based reasoning (to solve the problem directly and also to select
appropriate heuristics), multi-criteria methods and hyper-heuristics can be seen in [Burke and
Petrovic, 2002]. The range of methods and techniques that have been used over the years to tackle
exam timetabling problems have, on the whole, tended to draw on problem specific information and
particular heuristics. This paper deals with the goal of trying to develop more general exam timetabling
systems that are, to a certain extent, independent of the heuristics being employed.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
8
2 The Examination Timetabling Problem and Heuristic Ordering
Methods
2.1 Problem Definition
The general problem that will be presented and discussed in this paper is concerned with scheduling a
number of exams into a set number of periods or timeslots. This problem description was presented by
the authors in [Burke and Newall, 1999] but it is presented here again to aid the reader. The constraints
of the problem are:
1. No student should be required to attend two exams at the same time.
2. The allocated number of seats per period should not be exceeded.
3. Students should be allowed a reasonable study time between exams where possible.
In reality there are often a number of additional side constraints for certain exams, such as limiting an
exam to particular times or rooms etc.
To describe the problem more formally there are E exams that must be scheduled in P periods with S
examination seats available for each period (where E, P, and S are non-negative integers). There can be
multiple periods in any single day and it is possible that two of the exams could conflict with each
other. We use the term conflict between two exams to mean that at least one student is enrolled for both
exams. We consider two types of conflicts:
• First-Order Conflicts describe situations where conflicting exams are scheduled in the same
period. This is highly undesirable as it involves quarantining some students after one exam so
they may sit for the other exam after the main sitting. In all but the most difficult of situations
this is regarded as a basic hard constraint.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
9
• Second-Order Conflicts, on the other hand, are less important and represent situations where
two conflicting exams are not scheduled in the same period, but are scheduled in periods too
near to each other. For instance we might not want students to have to sit for two exams in consecutive periods, or to sit for two exams in the same day. The satisfaction of all of these constraints is often not practical and they are, therefore, usually treated as soft constraints.
Another important goal is to respect the limitations on available seating for each period. The following
notation is employed:
tip = 1
if exam i is scheduled in period p, 0 otherwise
c ij
is the number of students taking both exams i and j
d pq
is 3 if period p is on the same day as period q, 1 if they are on adjacent days and 0
otherwise. This provides a higher weight for same day conflicts because it is more
important to satisfy them.
si
is the number of students taking exam i
In some cases it may not be possible to construct a feasible timetable. Therefore we use an extra period,
the (P+1)th period, into which any exams that cannot be scheduled in a valid period can be placed.
There are no hard constraints to prevent scheduling in this period, but it should be heavily penalised.
In our experiment we use two alternative methods of solution evaluation in order to enable comparison
with a wider range of existing methods. These are shown in expressions (1) and (2). Expression (1)
represents a mechanism for measuring student spread based on proximity. Expression (2) represents an
alternative measure that sums up all cases where students must attend two exams in consecutive periods and strongly discourages incomplete timetables by weighting the number of unscheduled exams by
5000. If two consecutive periods are on the same day then any adjacent conflicts are weighted by three.
If there is a single night between them then the conflicts are only weighted by one, otherwise any conflicts are ignored (for example when periods are split by a weekend).
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
10
E–1
E
P–5
P
∑ ∑ ∑
E
∑
2
5–q+p
t ip t jq c ij +
i = 1 j =i p = 1 q = p+1
E–1
E
∑ ∑
∑ 50000ti ( P + 1 )
(1)
i=1
P
E
∑ tip tj ( p+1 ) cij dp ( p+1) +
t ip t j ( p – 1 ) c ij d p ( p – 1) +
i = 1 j = i+1 p = 1
∑ 5000ti ( P + 1 )
(2)
i=1
Regardless of the method of evaluation that is used the hard constraints that must be observed remain
the same and are presented in formulae (3) to (5). Equation (3) states that every event should be scheduled once, and only once in the timetable. Equation (4) specifies that no conflicting events should be
scheduled within the same period and the relation presented in (5) enforces the condition that the total
number of seats required for any period is not greater than the number of seats available.
P+1
∑ tip
= 1,
∀i ∈ { 1, …, E }
(3)
p=1
E–1
E
P
∑ ∑ ∑ tip tjp cij
= 0
(4)
∀p ∈ { 1, …, P }
(5)
i = 1 j = i+1 p = 1
E
∑ tip si ≤ S,
i=1
2.2 Heuristic Ordering Methods
As has been mentioned above, it is often useful to think of the clash elimination part of this problem in
terms of graph colouring. The relationship with the graph colouring problem can be illustrated by associating exams with vertices and by associating clashes between exams with the edges between the vertices. The relationship is completed by associating colours with the periods in the timetable. With this
obvious similarity it is no surprise that a number of graph colouring approaches (as briefly discussed in
the introduction above) have been applied to the exam timetable problem. Probably the best known of
these are Heuristic Ordering based methods [Carter et al., 1994,Carter and Laporte, 1996].
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
11
In terms of graph colouring, these methods function by using a heuristic to estimate how difficult each
vertex will be to colour. The overall idea is that by colouring the most difficult vertices first it is likely
that we will require a lower number of colours to completely colour the graph than if we had coloured
the vertices in a random order. Examples of common heuristics that are often used to estimate difficulty
include:
• Largest Degree First. Vertices with a large number of edges with other vertices (this being
termed the Degree) are dealt with first. In timetabling terms, these are considered to be more difficult to schedule than those with a lower degree.
• Largest Colour Degree First. This is similar to largest degree except that we only count edges
with vertices that are already coloured. For example, if we had a vertex with a high degree but
where none of its adjacent vertices have been coloured yet, then this is considered to be not currently difficult as it could be assigned any colour.
• Saturation Degree. This heuristic chooses the exams with the least valid periods that are currently available in the timetable [Brelaz, 1979]. The idea here is that these exams need scheduling sooner rather than later because the number of options could easily be exhausted.
As mentioned above in the introduction, a method was proposed by Burke, Newall and Weare [Burke
et al., 1998b] to improve on the results of traditional heuristic ordering methods by adding a random
element to the process. Here, instead of always scheduling the most “difficult” exam at each stage, we
could schedule a random choice of the five most difficult exams. With this random element we are
likely to get a different solution each time the algorithm is run. This has the effect that the method can
be run multiple times and the best solution can be kept. The results presented in [Burke et al., 1998b]
showed that this iterated approach could produce substantially better results that a single non-randomised run.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
12
3 An Adaptive Heuristic Construction Method
3.1 Motivation
A key motivation in the design of the method presented in this paper is the goal of developing more
general solutions to exam timetabling problems. The wealth of timetabling research over the years has
resulted in various exam timetabling algorithms/approaches and this work has certainly increased our
understanding of exam timetabling problems and how to solve them. However, it is the case that many
of the techniques and methods focus on one problem or type of problem. Carter and Laporte [Carter
and Laporte, 1996] point this out in their survey paper. Until recently, there has only been limited
research attempting to develop timetabling systems that can operate at a higher level of generality. This
situation is beginning to change (see [Burke et al., 2000, Burke and Petrovic, 2002, Ross et al., 1998,
Terrashima-Marin et al., 1999]). The overall goal of this paper is to attempt to make the choice of heuristic in the automated exam timetabling process much less important than it currently is.
As an example, consider the largest degree first heuristic. It functions by estimating how “difficult”
each vertex will be to colour. It is based on the assumption that vertices with more edges are typically
more “difficult” to colour. As we move from graph colouring to related problems like timetabling,
more additional factors come in to play. For instance, an event may be difficult because there is only
one suitable room available, or that it must be scheduled within a limited timespan. This means that
largest degree first becomes less useful as more factors are added to the problem.
In essence we want to arrive at an estimation of difficulty by experience, rather than by a pre-specified
metric. A good current example of this is saturation degree sorting, which estimates difficulty based on
the number of valid colours/periods currently available to assign to a vertex/event. Carter et al. conclude in [Carter et al., 1996] when comparing different algorithmic strategies:
“...We have shown that the combined use of a backtracking strategy and of a saturation degree
sorting rule almost systematically yields shorter schedules in less computing time. This is clearly
the case of the random test problems and for the more difficult real problems...”.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
13
We would expect experience based approaches to have a better worst and average case performance
than those based on pre-defined metrics, due to their ability to adapt to extreme circumstances. Solutions generated by pre-defined metrics can require more time consuming “tweaking” by a human operator when exceptions to the pre-defined metric arise.
Other examples of approaches that attempt to harness run-time experience include the concept of
Squeaky wheel optimisation proposed by Joslin and Clements [Joslin and Clements, 1999]. Here a
greedy constructor is applied to a problem, followed by an analysis phase that identifies problematic
elements in the produced solution. A prioritiser then ensures that the greedy constructor concentrates
more on these problematic elements next time, or as the authors phrase it, “The squeaky wheel gets the
grease”. This cycle is iterated until some stopping criteria are met. Selman and Kautz propose a similar
modification to their GSAT procedure [Selman and Kautz, 1993]. The GSAT procedure is a randomised local search procedure for solving propositional satisfiability problems. It functions by iteratively generating truth assignments and then successively “flips” the variable that leads the greatest
increase in clauses satisfied, in a steepest descent style. The proposed modification increases the
“weight” of clauses if they are still unsatisfied at the end of the local search procedure. This has the
effect that in subsequent attempts the local search element will concentrate more on satisfying these
clauses.
In [Burke et al., 1998b] an iterated heuristic ordering method was used. However, each iteration of this
method was essentially isolated from every other iteration with no reuse of partial solutions. It seems
logical that, instead of merely using a heuristic to guess how difficult an exam will be to schedule, we
could augment this with the experiences of previous iterations when trying to schedule that exam.
As an example, suppose that exam A is deemed by our heuristic to be only moderately difficult. The
algorithm will then schedule all the exams that are deemed to be more difficult before it comes to exam
A. By this time, it may not be able to schedule exam A without breaking one or more hard constraints.
If this is the case, it would a good idea to promote exam A up the difficulty order in future runs of the
algorithm. This could be thought of as being analogous to a human performing the task manually. If he/
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
14
she has trouble scheduling certain exams then he/she will probably remember this for the following
year and give them priority. The proposed approach is motivated by these observations.
It could also be argued that such experience based approaches could be employed to reduce violations
of soft constraints. Of course, this is somewhat problematic, as determining what is an acceptable level
of violation is a very subjective matter. We will, however, attempt to address this issue in our implementation which is presented in the next section.
3.2 Algorithms
3.2.1 The Basic Heuristic Approach
Figure 1 shows sample pseudo-code of the simple heuristic method used as the basis for these experiments, where difficulty(e) is given by the chosen heuristic. Quite often the heuristic chosen is not completely appropriate. Previously scheduled exams may need to be moved or unscheduled [Carter
et al., 1996], in order to produce a feasible timetable.
3.2.2 Heuristic Modifiers
To enable the ordering to be adapted over time we introduce the idea of a heuristic modifier. This is
simply an integer that is added to that produced by our heuristic to give a new measure of difficulty that
varies over time. The perceived difficulty of an exam e at iteration i is presented in equation (6), where
heuristic(e) gives the heuristic estimate of difficulty and heurmodei gives the heuristic modifier for
exam e at iteration i.
difficulty ( e, i ) = heuristic ( e ) + heurmod ei
(6)
Having defined this modifier we must now determine the circumstances under which we should alter it.
One possibility would be to increase the modifier if we find that an exam cannot be scheduled without
breaking one or more hard constraints. This should help in the generation of feasible timetables, which
can itself be a problem. In addition, we could also increase the modifier if an exam cannot be scheduled
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
15
without breaking certain soft constraints, the aim being to improve the overall quality of the timetable.
In this study both approaches will be investigated.
This process is analogous to squeaky wheel optimisation [Joslin and Clements, 1999]. In terms of this,
the basic heuristic ordering is our greedy algorithm, exams that cannot be scheduled easily are the
“squeaky wheels” and the heuristic modifier represents the element of prioritisation.
3.2.3 Considering hard Constraints only
As we are concerned (in this section) with hard constraints only we need only increase the modifier
when an exam cannot be scheduled without breaking any such constraints. Formulation (7) shows how
to increase the modifier for the next iteration, where the function modify represents the strategy being
used to increase the modifier.
 modify ( heurmod ei )
heurmod ei + 1 = 
 heurmod ei
, iff exam e cannot be scheduled
, otherwise
(7)
A choice of incrementing by 1 ( modify(x) = x + 1 ) represents the finest grade of adjustment, at least
with an integer based system. It is possible to use higher values. It is also possible to employ alternative
strategies such as multiplying by a value greater than 1, but we would risk losing any benefit from the
initial ordering. In general, the higher this value, the less relevant the initial ordering will be. The disadvantage of using a fine adjustment level is that it will generally take more iterations to make any
large adjustments that may be necessary.
We will experiment with two other strategies for adapting the modifier. Firstly, we will consider an
exponent based approached where the modifier is incremented by a value of 2n when n is the number
of times exam e’s modifier has already been increased. For example, the first time an exam’s modifier
is increased it will be by 1, the 2nd time it will be increased by 2, the 3rd time by 4 and so on. We
would expect this to increase the speed of the approach when considering poor initial orderings, due to
the larger steps involved. The second method is to increment the modifier by some random value. By
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
16
introducing this element of randomness we may open up options not available in the deterministic
algorithm.
It is clear that implementing this will require only a negligible amount of system resources over the
standard approach. The pseudo-code for the method, using the definitions presented in (6) and (7), is
shown in figure 2.
3.2.4 Considering both hard and soft Constraints
As mentioned above, considering soft constraints presents the question of “What level of violation is
acceptable?”. This will inevitably depend on the nature of each problem and will vary considerably
from institution to institution. For this approach we choose to implement a method whereby the level of
acceptable violation is lowered gradually under certain conditions.
In all test cases we use a linear weighted penalty function to evaluate solutions. This can also be used to
evaluate the cost of scheduling a single exam. We can then say that an exam’s cost is acceptable if it is
less than a variable maxCost. A reasonable initial value for maxCost can be found by carrying out a
dummy run and selecting the highest cost found when scheduling an exam. This can then be gradually
reduced in order to increase the pressure to find better solutions.
The rate at which to reduce maxCost presents a new problem. To avoid potential pitfalls, maxCost is
only reduced when the solution is in a state of equilibrium. By equilibrium we mean that all exams can
be scheduled with a cost lower than maxCost. Once the solution is in such a state, subsequent runs will
produce identical results until maxCost is lowered enough to cause at least one exam to be viewed as
unacceptable.
It is possible that, when using this method, we may have cases where two or more exams cannot all
have a cost lower than maxCost at the same time, even when they are all scheduled early in the process.
This results in a cycle that, unless broken, will continue forever. To break these cycles we maintain a
count of how many times an exam has had its modifier increased. Once it reaches a set limit (of say 100
times) we then regard an exam as being exempt. What this essentially means is that a solution can still
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
17
be in a state of equilibrium even if one or more exempt exams have just been modified. This allows the
algorithm to lower maxCost, even when a cycle occurs, and concentrate on the other exams. The algorithm for this process is shown in pseudo-code in figure 3, where maxModify represents the number of
time an exam’s modifier can be increased before it is ignored (100 times for our experiments).
A possible advantage of this approach is that the maximum cost for any exam, at least for solutions in a
state of equilibrium, will tend to be lower because we have a set limit on maxCost. This might be
viewed as being a “fairer” system. The system of maxCost could also be adapted for different needs.
For instance, we could determine it in terms of “cost per student” so large exams take more of the burden from the smaller exams. It also has possibilities for multi-objective problems where we have a
hierarchy of constraints. For example we might want to try and eliminate students having two exams in
a row, before trying to eliminate two exams in a day.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
18
4 Experimental Results
4.1 Benchmark Setup
The method has been tested on a wide range of real world timetabling benchmark problems that were
used by Carter et al. [Carter et al., 1996] to evaluate algorithmic strategies for timetabling. These problems were also used by Di Gaspero and Schaerf [Di Gaspero and Schaerf, 2001] to test tabu search
techniques. A subset of these problems was also used by Burke and Newall[Burke and Newall, 1999]
to evaluate the effects of decomposing a problem into smaller chunks. Thanks to this use of freely
available data we can make direct comparisons with these three different techniques. Table 1 lists all
problems used for testing together with their characteristics. Where appropriate the number of periods
used for testing by Carter et al. is given.
Two methods were used for evaluating solution quality. The first is that shown by Carter et al. [Carter
et al., 1996] where a proximity cost ws is assigned whenever a student has to sit two examinations s
periods apart. These weights are defined as: w1 = 16, w2 = 8, w3 = 4, w4 = 2, w5 = 1. The sum of all
these measures is then divided by the number of students to give an average penalty for each student.
For these problems there are no limits on the amount of seating available during each period. In the
context of our earlier problem definition this involves minimising expression (1) and ignoring the seating limitation requirement of expression (5).
The second method of evaluation is that presented by Burke and Newall [Burke and Newall, 1999]. For
these problems, there is a set limit on the number of students that can sit an exam in each period. The
penalty function is calculated by counting all situations where a student has two exams in adjacent
periods on the same day, weighted by 3, plus the number of situations where a student has two exams
in adjacent periods but split over 2 days. Any exams that are left unscheduled incur a further penalty of
5000 each. The weights were arbitrarily chosen though they are intended to reflect the relative importance of the constraints. In the context of our problem definition, we would minimise expression (2),
subject to all the constraints.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
19
A major aspect of this proposed approach is the ability to adapt initial choices of heuristic. Experiments
were conducted with various initial orderings for the method. In the case of a tie, we simply employ the
order in which the data has been input. The initial orderings we used are:
• Largest Degree First (LD). This is generally thought of as a simple but reasonably effective
heuristic for graph colouring and timetabling problems. We might, therefore, expect our adaptive approach to work reasonably well when started on this heuristic.
• Flat Ordering. Here, we start the process by assigning the value of zero to each exam instead of
using the degree, resulting in a flat ordering. We would expect poor results when using this
ordering and, therefore, there is a lot of scope for the adaptive method to outperform it.
• Smallest Degree First (SmD). Here we take the opposite of Largest degree first to form a “misleading” heuristic. We might expect results to be even worse than when using a randomised or
flat ordering. The challenge here for an adaptive approach is that it will have to virtually reverse
the ordering before it will find reasonable results.
• Saturation Degree(SatD). This typically produces a much better ordering than largest degree
first as reported in [Carter et al., 1996]. Due to its higher effectiveness we can produce a better
quality starting point, and investigate whether this leads to a higher quality final solution. As saturation degree is a dynamic ordering it is only used to create an initial ordering.
• Random Orderings. Here of course we start the algorithm on randomly generated orderings.
The aim here is to explore the situation when dealing with a chaotic starting point.
Experiments were performed both with and without the mechanism for taking into account soft constraints for modification. As the method generally does not include any random elements only a single
run for each scenario and problem needed to be run in most cases. Unless otherwise specified the
method was run for a maximum of 2,000 iterations, with the time reported being the time taken to find
the best solution.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
20
All the experiments presented here were run on an Athlon 900Mhz system running linux. Please note
that the other methods in the comparisons were run on slower systems. The results given by Di Gaspero and Schaerf [Di Gaspero and Schaerf, 2001] were performed on an Athlon 650Mhz, those of
Burke and Newall [Burke and Newall, 1999] were run on a 300Mhz Alpha EV-6 system. The experiments presented by Carter et al. [Carter et al., 1996] were run on a 16.7Mhz SUN SPARC sever 330.
Given the relatively slow speed of this machine, cpu times were not included with the comparisons for
the experiments of Carter et al. Where we compare with randomised approaches we take the average
costs, as this seems the fairest way to compare randomised and deterministic methods.
4.2 When Adapting Based on Hard Constraints Only
The first experiments were performed using the mechanism to increase the modifier only when an
exam could not be scheduled due to hard constraints. Here the method will stabilise and then terminate
when it has found a feasible solution. As it does not particularly try to optimise the soft constraints we
would not expect it to excel with respect to those constraints.
4.2.1 Using a Largest Degree First Initial Ordering
Table 2 shows the results obtained when using Largest Degree First as our base heuristic. The best
results in any particular table are presented in bold. For each table, a row showing the totals of the columns is provided to give an aggregate impression of performance over all instances. Even though the
algorithm does not modify on the basis of soft constraints it manages to achieve the best result on the
car-f-92 and car-s-91 problems, and is comparable to Carter et al.’s approach in the other cases. It
should be noted that the results of Carter et al. represent a range based on using different heuristics.
The best result was not always found with the same heuristic and, of course, our approach employs the
same heuristic throughout. The Tabu Solver also competes well in most cases.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
21
4.2.2 Using other Initial Orderings
In order to establish how much the initial ordering can affect the results, experiments were conducted
with other ordering methods. Firstly, a flat ordering was used to identify how much benefit there is in
using an initial heuristic ordering at all. Secondly the reverse of largest degree (i.e. smallest degree
first) was tried so as to evaluate the performance of the method when given an inappropriate heuristic.
Saturation degree was used to investigate the effects with a good initial ordering and random orderings
were used to investigate performance in more chaotic circumstances.
When using a flat initial ordering, we lose any benefits gained by using largest degree first. We might
expect our method here to maybe generate worse solutions and to take longer as it has to evolve its own
ordering. The results in table 3 show that in most cases we do indeed achieve a slightly worse result.
However, the time taken is generally considerably less. Here, the results for car-f-92, sta-f-83, tre-s-92
and uta-s-93 are actually slightly better than when using largest degree first. This casts doubt on how
useful largest degree first is within the adaptive method.
As we expected, saturation degree performs effectively, obtaining the best result from all of the orderings in six of the test cases. Recall that the bold figures represent the best of the results presented in the
table. It is worth noting that adaptation of the ordering was only required for the hec-s-93 problem in
order to obtain a feasible solution. Even with no adaption, however, it was still typically no faster than
say the flat approach due to the increased overheads of calculating the saturation degree. It is interesting to see how the situation develops when we start considering soft constraints in the adaptation process.
As largest degree first is generally regarded to be a reasonable heuristic for timetable problems we
would expect the reverse of this ordering to achieve very poor results when used on its own. Therefore,
starting the method with an initial smallest degree first method should present a challenge to the
method and give a good idea of worst case performance. The ordering was achieved here by giving
each exam a heuristic value obtained by subtracting the degree of the exam from the total number of
exams. The results in table 3 show that, as we would expect, it takes much longer for the algorithm to
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
22
evolve an ordering that produces a feasible solution. This is because the genuinely “hard” exams will
tend to be placed towards the end of the ordering. There is then a long process where they must fight
their way towards the front. However, we found that, with the flat initial ordering, there are cases
where the final result is actually better than when using largest degree first. In the case of ute-s-92, it
produces the best results across the three orderings although it does not beat the other approaches (by
Carter et al. and Di Gaspero and Schaerf). In all cases though, the results remain competitive, which is
a considerable achievement given that the method has been given an inappropriate heuristic (as illustrated by the amount of adaption required to reach feasibility). It should be noted that even the tabu
solver of Di Gaspero and Schaerf uses graph colouring heuristics to generate an initial feasible solution.
When using a random based ordering, we assign an initial random priority in the range [0, N] to each
exam. As these experiments (unlike the previous ones) incorporate a random element, all runs were
performed 5 times with varying random seeds. An interesting observation here is that the value of N
has a big impact on the performance of the method, resulting in run times far in excess of those
required even by smallest degree first. The reason for this lies in the way we calculate the initial values
for smallest degree (shown above) which results in a smaller spread of values when compared to a random distribution. Table 4 shows the results when we set N as the number of exams in the problem and
again with half the number of exams. Note that while, in both cases, the method achieves roughly
equivalent final quality it takes substantially longer on average for the larger random range. The
amount of time taken here is inherently linked to the size of the random range used and the amount we
increment the heuristic modifier by. A smaller random range, or a larger modifier, will tend to be more
like a flat ordering.
4.2.3 Using Exponent Based Adaption
As an alternative to simply incrementing the heuristic modifier by 1 each time there is a failure, it is
worthwhile investigating the more drastic approach of exponent based adaption as described in section
3.2.3. We would expect this to be especially useful on poor initial orderings to accelerate the process.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
23
For these experiments, there is no point in using flat initial orderings, as it would merely reproduce the
behaviour when using incremental adaption. As initial orderings with saturation degree rarely require
any adaption at all, these too were excluded from these experiments. Table 5 shows the result of using
this exponent based approach on the applicable initial orderings. While the quality of produced solutions remains largely the same, with just minor variations, the time taken to produce a feasible solution
is drastically reduced. This effect is most noticeable when using a random initial ordering where a feasible solution is found in approximately a tenth of the time it takes when using simple incremental
adaption. This dramatic speed up is illustrated in figure 4 for the car-f-92 problem.
4.3 When Adapting Based on both hard and soft Constraints
As we have illustrated, competitive results were found when adapting based on hard constraint violation. We would expect it to be the case that we can improve the results further by considering violation
of soft constraints using the methods out-lined above. The results shown in table 6 confirm this to be the
case. Using the LD initial ordering, the approach is able to produce better results on 4 of the test cases
than the Tabu solver and heuristic approaches presented in table 2, even though the initial aim of the
approach is to be general and competitive rather than to “beat” other approaches to the problem.
Indeed, we would have considered the approach to be a success if it could get into the same “ball-park”
as these heuristic dependent approaches, while being more general, much quicker and much easier to
implement. It is not only comparable to these approaches, but can actually produce better results in
some cases. An interesting observation here is that saturation degree no longer achieves the best final
result in the majority of cases, showing that the adaptive method does in fact compensate for bad orderings. This is illustrated in figure 5, which shows the performance of Largest degree and Saturation
degree over time on the car-f-02 problem. While saturation degree produces a better result in the very
early stages it is soon caught up by the process when started with largest degree.
As before, we shall present experiments that employ exponent based adaption. Table 7 shows the
results when started with largest degree, saturation degree and smallest degree. The final results show
some minor differences but there is no overall improvement. The only benefit, in this context, appears
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
24
to be the ability to reach feasible solutions earlier on in the process when started with poor orderings.
This is illustrated in figure 6, showing the performance of incremental and exponential adaption on the
car-f-92 problem when using smallest degree first. This behaviour can be explained by the large scale
modification tending to make the initial ordering relatively insignificant.
A final set of experiments was carried out using a random modifier (between 1 and 8) with an initial
saturation degree ordering. In this case, due to the random element, each experiment was run 5 times
with average, worst and best results taken. These are shown in table 8. Some modest improvements are
achieved on average when compared with the deterministic methods of adaption.
It is also possible that, when using this random based adaption, there is more scope for improvement
given a longer maximum run time. To test this out the experiment was repeated using a maximum of
20,000 iterations. The results, presented in table 9, show that some further modest improvements can
be made in some cases if the user is prepared to wait a little longer.
Some brief experiments were also performed using a backtracking operator similar to that used by
Carter et al. [Carter et al., 1996]. While some problems received a marginal improvement, many problems in fact achieved worse results. This is almost certainly due to the backtracking procedure frustrating the process of adaption. Our conclusion is that backtracking adds nothing to the adaptive approach.
4.4 Comparison of Best and Worst Cases
As mentioned earlier, we would expect experienced based approaches to have a better average and
worst case performance. As we have already investigated average performances, it would be interesting to compare best and worst case performance across our available data. This information is presented in table 10, with best overall results marked in bold and worst overall results marked in italics. It
should be noted that, as the worst case results are not published for the experiments of di Gaspero and
Schaerf [Di Gaspero and Schaerf, 2001], we cannot attribute the worst overall case to this method
unless the average itself is worse than the other methods. It may be that, in reality, this method produces worst cases not indicated here. It should also be noted that we include the cases for the adaptive
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
25
method across all experiments when considering both hard and soft constraints. The adaptive method
comes out well in this comparison, attaining the best overall results in 6 of the 11 test cases and the
worse result in only 1, although this may have been different if the worst case figures for the Tabu
solver were available. It is quite notable that the adaptive results fall within a narrow range, to the
extent that the worst result from the adaptive approach is often superior to the best result found by one
or both of the other methods.
Focussing on the adaptive results, it is worth noting that the best results for 10 of the problems were
found using the randomised adaption with extended run times which, of course, leads to the longer run
times listed in the table. This configuration also contributed only 1 worst case which suggests that it is
quite robust.
The complete results presented in this paper, together with final solutions can be downloaded from:
http://www.asap.cs.nott.ac.uk/misc/jpn/adaptive/
4.5 Comparisons with Burke and Newall
In addition to the above comparisons, we shall also compare with the approach presented by Burke and
Newall [Burke and Newall, 1999]. Results for the same data sets were also presented by di Gaspero
and Schaerf [Di Gaspero and Schaerf, 2001] providing another angle for comparison with the Tabu
Solver approach. In general, the problems here are based more on the real life situation and include
limits on the number of seats per period. The penalty function used was as given in expression (2).
Table 11 lists the problems used and the time and seating limitations placed upon them.
As the approach presented in [Burke and Newall, 1999] involves a number of different techniques and
parameters we will compare with the setup that produced the best results in general, namely saturation
degree with a set size of 50 and a look ahead set. Table 12 shows the comparative results of Di Gaspero
and Schaerf’s tabu search solver, Burke and Newall’s decomposition approach [Burke and
Newall, 1999] and the adaptive method with largest degree first, saturation degree and flat initial orderings. As can be seen the decomposition approach dominates the results for these problems. The adap-
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
26
tive approach, however, does follow closely behind and has substantially lower run times, even when
taking the difference in machine speed into account. The decomposition approach is also a much more
complicated method to implement. As the the randomised adaption with an initial saturation degree did
well in the previous experiments we also applied this to these problems. Table 13 shows that this produces results comparable to the decomposition method in a comparable amount of time, with an exception being the huge pur-s-93 problem on which the decomposition method seems to excel.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
27
5 Conclusions
The experiments leave two questions outstanding: “Which initial orderings should we use?” and
“Which adaption method is best?” Unfortunately these two questions are interrelated so there is no definite answer to either question. Any answers are probably best illustrated in the context of three varying
situations:
• A reasonable ordering heuristic exists and is known. In this case, it makes perfect sense to use
this heuristic as our initial ordering, combined with incremental adaption to tweak the solution
further. The saturation degree ordering in our experiments is a good example of this case. It performs well on our test problems in itself. However solution quality can be improved further
through the adaption mechanism.
• No effective ordering heuristic is known. Here it is probably wise to start with a flat initial
ordering. The choice of adaptation strategy on a flat ordering is largely irrelevant (random adaption would be the only one to produce varying results). The time taken to generate solutions may
be longer but equivalent quality solutions should still be attainable.
• An ordering heuristic is known but its efficacy on all problems of interest varies. This situation
would lie between the other two extreme cases and is likely to be the one that arises most in
practice. Here, it may be worthwhile to use the heuristic, but combined with exponential adaption, which has been shown to overcome even extremely bad initial orderings relativity quickly.
In addition to the above points, randomised adaption also shows great promise to improve on results
further, especially if the user is willing to wait a little longer for results and/or run the method multiple
times.
Overall, the experiments have shown the adaptive method to perform very efficiently and competitively on a wide range of problems. The method compares well with other more time consuming
approaches and is relatively easy to implement. A very positive feature of the method presented here is
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
28
the narrow gap between the best and worst results over the extensive range of experiments. This demonstrates a certain robustness, regardless of which possible configuration is chosen.
The proposed method can be thought of as giving us two mechanisms by which to solve the problem. If
the given ordering heuristic is ideal for the problem, then a good solution will be produced at the start
of the process. Otherwise if the heuristic is not, as is likely, entirely suitable then the adaptive method
can try to improve or salvage the initial ordering. This could lead to significant developments in automated timetabling systems that would, ultimately, be used by relatively novice users with little familiarity with graph theory and little idea of the suitability or unsuitability of particular heuristics. Another
potential benefit is that more general “knowledge poor” systems are cheaper to develop than problem
specific systems and this could lead to greater use of more general systems in the user community.
While numerical results on relatively simple problems, like examination timetabling, are an important
indication of effectiveness, a major advantage of this approach is the fundamentally more general way
in which it operates. The approach could potentially be investigated for problems other than exam
timetabling where similar ordering based construction heuristics exist. Another possible advantage of
this approach may be its ability to gradually increase the pressure to produce “fairer” timetables. While
the approach we have taken is to cap the total penalty of any one exam, it would be possible to concentrate only on certain constraints that we regard to be particularly important. For instance, in exam timetabling it is usually very desirable to spread the exams as evenly as possible across the entire timetable.
Therefore, we could have varying levels of spread, such as having 1 full period between exams, then 2
full periods between exams etc. We could call these constraints “firm” in that they are actually treated
as lying somewhere between hard and soft constraints.
Acknowledgments
The research described in this paper was supported by EPSRC grant GR/N36837/01.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
29
References
Bardadym, V. A. (1996). Computer-aided school and university timetabling: A new wave. In
Edmund Burke, P. R., editor, The Practice and Theory of Automated Timetabling: Selected
Papers from the 1st International Conference, Lecture Notes in Computer Science 1153, pages
22–45. Springer-Verlag, Berlin.
Boizumault, P., Delon, Y., and Peridy, L. (1996). Constraint logic programming for examination
timetabling. Journal of Logic Programming, 26(2):217–233.
Boufflet, J. P. and Negre, S. (1996). Three methods used to solve an examination timetable problem. In
Burke, E. and Ross, P., editors, The Practice and Theory of Automated Timetabling: Selected
Papers from the 1st International Conference, Lecture Notes in Computer Science 1153, pages
327–344. Springer-Verlag, Berlin.
Brelaz, D. (1979). New methods to colour the vertices of a graph. Communications of the Association
for Computing Machinery, 22:251–256.
Bullnheimer, B. (1998). An examination scheduling model to maximize students’ study time. In
Edmund K. Burke, M. W. C., editor, Lecture Notes in Computer Science 1408, pages 78–91.
Springer-Verlag, Berlin.
Burke, E., Bykov, Y., and Petrovic, S. (2001). A mulitcriteria approach to timetabling. In Burke, E. K.
and Erben, W., editors, Practice and Theory of Automated Timetabling III (Lecture Notes in
Computer Science 2079), pages 118–131. Springer, Berlin.
Burke, E. and Petrovic, S. (2002). Recent research directions in automated timetabling. To appear in the
European Journal of Operations Research.
Burke, E. K., Elliman, D. G., Ford, P. H., and Weare, R. F. (1996a). Examination timetabling in british
universities - a survey. In Burke, E. and Ross, P., editors, The Practice and Theory of Automated
Timetabling: Selected Papers from the 1st International Conference, Lecture Notes in Computer
Science 1153, pages 76–90. Springer-Verlag, Berlin.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
30
Burke, E. K., Elliman, D. G., and Weare, R. F. (1994). A university timetabling system based on graph
colouring and constraint manipulation. Journal of Research on Computing in Education, 27(1):1–
18.
Burke, E. K., Elliman, D. G., and Weare, R. F. (1995a). A hybrid genetic algorithm for highly
constrained timetabling problems. In Eshelman, L. J., editor, Genetic Algorithms: Proceedings of
the 6th Internation Conference, pages 605–610, San Francisco. Morgan Kaufmann.
Burke, E. K., Elliman, D. G., and Weare, R. F. (1995b). Specialised recombinative operators for
timetabling problems. In Fogarty, T. C., editor, Lecture Notes in Computer Science 993 (AISB
Workshop on Evolutionary Computing), pages 75–85. Springer-Verlag, Berlin.
Burke, E. K., Jackson, K., Kingston, J. H., and Weare, R. F. (1997). Automated university timetabling:
The state of the art. The Computer Journal, 40(9):565–571.
Burke, E. K. and Newall, J. P. (1999). A multi-stage evolutionary algorithm for the timetable problem.
IEEE Transactions on Evolutionary Computation, 3(1):63–74.
Burke, E. K., Newall, J. P., and Weare, R. F. (1996b). A memetic algorithm for university exam
timetabling. In Burke, E. and Ross, P., editors, The Practice and Theory of Automated
Timetabling: Selected Papers from the 1st International Conference, Lecture Notes in Computer
Science 1153, pages 241–250. Springer-Verlag, Berlin.
Burke, E. K., Newall, J. P., and Weare, R. F. (1998a). Initialization strategies and diversity in
evolutionary timetabling. Evolutionary Computation, 6(1):81–103.
Burke, E. K., Newall, J. P., and Weare, R. F. (1998b). A simple heuristically guided search for the
timetable problem. In International ICSC Symposium on Engineering of Intelligent Systems (EIS
’98), pages 574–579. ICSC Academic Press, Canada/Switzerland.
Burke, E. K., Ross, P. M., Cowling, P. I., Petrovic, S., and Kendall, G. (2000). An investigation of
hyper-heuristic methods. Research Project funded by the Engineering and Physical Sciences
Research Council (EPSRC), Grant Ref: GR/N36837/01.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
31
Carter, M. W. (1986). A survey of practical applications of examination timetabling. Operations
Research, 34:193–202.
Carter, M. W. and Johnson, D. G. (2001). Examination timetabling: Algorithmic strategies and
applications. Extended Clique Initialization in Examination Timetabling, 52(5):538–544.
Carter, M. W. and Laporte, G. (1996). Recent developments in practical examination timetabling. In
Burke, E. and Ross, P., editors, The Practice and Theory of Automated Timetabling: Selected
Papers from the 1st International Conference, Lecture Notes in Computer Science 1153, pages
3–21. Springer-Verlag, Berlin.
Carter, M. W., Laporte, G., and J.W.Chinneck (1994). A general examination scheduling system.
Interfaces, 11:109–120.
Carter, M. W., Laporte, G., and Lee, S. (1996). Examination timetabling: Algorithmic strategies and
applications. Journal of the Operational Research Society, 47(3):373–383.
Corne, D., Fang, H. L., and Mellish, C. (1993). Solving the modular exam scheduling problem with
genetic algorithms. In Proceedings of the Sixth International Conference on Industrial and
Engineering Aplications of Artificial Intelligence and Expert Systems, pages 370–373.
Corne, D. and Ross, P. (1996). Peckish initialisation strategies for evolutionary timetabling. In
Edmund Burke, P. R., editor, The Practice and Theory of Automated Timetabling: Selected
Papers from the 1st International Conference, Lecture Notes in Computer Science 1153, pages
227–240. Springer-Verlag, Berlin.
Corne, D., Ross, P., and Fang, H. (1994). Fast practical evolutionary timetabling. In T.C.Fogarty, editor,
Lecture Notes in Computer Science 865, pages 250–263. Springer-Verlag, Berlin.
David, P. (1998). A constraint-based approach for examination timetabling using local repair
techniques. In Burke, E. K. and Carter, M. W., editors, Lecture Notes in Computer Science 1408,
pages 169–186. Springer-Verlag, Berlin.
de Werra, D. (1985). An introduction to timetabling. European Journal of Operations Research,
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
32
19:151–162.
de Werra, D. (1997). Restricted coloring models for timetabling. DMATH: Discrete Mathematics, 165.
Di Gaspero, L. and Schaerf, A. (2001). Tabu search techniques for examination timetabling. In Burke,
E. K. and Erben, W., editors, Practice and Theory of Automated Timetabling III (Lecture Notes
in Computer Science 2079), pages 104–117. Springer, Berlin.
Dowsland, K. A. (1996). Simulated annealing solutions for multi-objective scheduling and timetabling.
In Modern Heuristic Search Methods, pages 155–166. Wiley, Chichester, England.
Dowsland, K. A. (1998). Off=the-peg or made-to-measure timetabling and scheduling with sa and ts. In
Burke, E. K. and Carter, M. W., editors, Lecture Notes in Computer Science 1408, pages 37–52.
Springer-Verlag, Berlin.
Erben, W. (2001). A grouping genetic algorithm for graph colouring and exam timetabling. In Burke,
E. K. and Erben, W., editors, Practice and Theory of Automated Timetabling III (Lecture Notes
in Computer Science 2079), pages 132–156. Springer, Berlin.
Ergul, A. (1996). GA-based examination scheduling experience at a middle east technical university. In
Edmund Burke, P. R., editor, The Practice and Theory of Automated Timetabling: Selected
Papers from the 1st International Conference, Lecture Notes in Computer Science 1153, pages
198–211. Springer-Verlag, Berlin.
Falkenauer, E. (1998). Genetic Algorithms and Grouping Problems. John Wiley and Sons, Chicester.
Foxley, E. and Lockyer, K. (1968). The construction of examination timetables by computer. The
Computer Journal, 11:264–268.
Hertz, A. (1991). Tabu search for large scale timetabling problems. European Journal of Operations
Research, 54:39–47.
Joslin, D. E. and Clements, D. P. (1999). Squeaky wheel optimization. Journal of Artificial Intelligence
Research, 10:353–373.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
33
Mehta, N. K. (1981). The application of a graph coloring method to an examination scheduling problem.
Interfaces, 11:57–64.
Nuijten, W. P. M., Kunnen, G. M., Aarts, E. H. L., and Dignum, F. P. M. (1994). Examination
timetabling: A case study for constraint satisfaction. In ECAI ’94 Workshop on Constraint
Satisfaction Issues Raised by Practical Applications, pages 11–19.
Ross, P. and Corne, D. (1995). Comparing genetic algorithms, simulated annealing and stochastic
hillclimbing on timetable problems. In Fogarty, T. C., editor, Lecture Notes in Computer Science
993 (AISB Workshop on Evolutionary Computing), pages 94–102. Springer-Verlag, Berlin.
Ross, P., Corne, D., and Fang, H.-L. (1994). Improving evolutionary timetabling with delta evaluation
and directed mutation. In Davidor, Y., Schwefel, H.-P., and Manner, R., editors, Parallel Problem
Solving in Nature, volume III, pages 565–566. Springer-Verlag, Berlin.
Ross, P., Hart, E., and Corne, D. (1998). Some observations about ga based timetabling. In Edmund
K. Burke, M. W. C., editor, Lecture Notes in Computer Science 1408, pages 115–130. SpringerVerlag, Berlin.
Schaerf, A. (1999). A survey of automated timetabling. Artificial Intelligence Review, 13(2):87–127.
Selman, B. and Kautz, H. (1993). Domain-independent extensions to gsat: Solving large structured
satisfiability problems. In Proceedings of the 13th Int’l Joint Conf. on Artificial Intelligence,
pages 290–295.
Terrashima-Marin, H., Ross, P., and Valenzuela-Rendon, M. (1999). Evolution of constraint
satisfaction strategies in examination timetabling. In Proceedings of the Genetic and
Evolutionary Computation Conference, pages 635–642. Morgan Kaufmann, San Francisco.
Thompson, J. M. and Dowsland, K. A. (1996a). General cooling schedules for a simulated annealing
based timetabling system. In Edmund Burke, P. R., editor, The Practice and Theory of Automated
Timetabling: Selected Papers from the 1st International Conference, Lecture Notes in Computer
Science 1153, pages 345–364. Springer-Verlag, Berlin.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
34
Thompson, J. M. and Dowsland, K. A. (1996b). Variants of simulated annealing for the examination
timetabling problem. Annals of Operations Research, (63):105–128.
Thompson, J. M. and Dowsland, K. A. (1998). A robust simulated annealing based examination
timetabling system. Computers and Operations Research, 25(7/8):637–648.
Welsh, D. J. A. and Powell, M. B. (1967). An upper bound for the chromatic number of a graph and its
application to timetable problems. The Computer Journal, 10:85–86.
White, G. M. and Xie, B. S. (2001). Examination timetables and tabu search with longer-term memory.
In Burke, E. K. and Erben, W., editors, Practice and Theory of Automated Timetabling III
(Lecture Notes in Computer Science 2079), pages 85–103. Springer, Berlin.
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
35
Tables
Density
Data
Institution
Periods
Number
Number
of
of
Exams
Students
Number of
of
Enrolments
Conflict
Matrix
car-f-92
Carleton University, Ottawa
32
543
18,419
55,552
0.14
car-s-91
Carleton University, Ottawa
35
682
16,925
56,877
0.13
24
190
1,125
8,108
0.29
18
81
2,823
10632
0.20
Earl Haig Collegiate Institute,
ear-f-83
Toronto
Ecole des Hautes Etudes
hec-s-93
Commerciales, Montreal
kfu-s-93
King Fahd University, Dharan
20
461
5,349
25,118
0.06
lse-f-91
London School of Economics
18
381
2,726
10919
0.06
pur-s-93
Purdue University, Indiana
43
2,419
30,032
120,690
0.03
13
139
611
5,751
0.14
23
261
4,360
14901
0.18
35
622
21,267
58,981
0.13
10
184
2,750
11,796
0.08
21
181
941
6029
0.27
N/A
800
7,896
34,265
0.03
St Andrew’s Junior High
sta-f-83
School, Toronto
Trent University, Peterborough,
tre-s-92
Ontario
Faculty of Arts and Sciences,
uta-s-93
University of Toronto
Faculty of Engineering,
ute-s-92
University of Toronto
York Mills Collegiate Institute,
yor-f-83
Toronto
nott
Nottingham University, UK
Table 1. The benchmark problems used
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
36
Tabu Solver (avg.)
Carter et al.
Data
Adaptive (LD)
Di Gaspero & Schaerf
Cost
Time
Cost
Cost
Time (s)
car-f-92
5.6
834.4
6.2-7.6
5.23
1.4
car-s-91
6.5
30.12
7.1-7.9
5.97
1.6
ear-f-83
46.7
3.6
36.4-46.5
40.53
0.3
hec-s-93
12.6
4.9
10.8-15.9
15.01
0.02
kfu-s-93
19.5
14.8
14.0-20.8
18.07
1.2
lse-f-91
15.9
23.7
10.5-13.1
13.8
0.4
sta-f-83
166.8
7.3
161.5-165.7
169.71
0.1
tre-s-92
10.5
14.5
9.6-11.0
10.63
0.3
uta-s-93
4.5
36.0
3.5-4.5
4.48
2.5
ute-s-92
31.3
16.0
25.8-38.3
33.82
0.1
yor-f-83
42.1
25.2
41.7-49.9
45.95
0.3
Total
362
1010.5
327.1-381.2
363.20
8.2
Table 2. Results when seeding with largest degree first
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
37
LD initial order
flat initial order
SmD initial order
SatD initial order
Cost
Time(s)
Cost
Time(s)
Cost
Time(s)
Cost
Time
car-f-92
5.23
1.4
4.92
0.9
6.00
7.9
4.86
0.9
car-s-91
5.97
1.6
6.06
1.15
6.91
10.8
5.52
1.7
ear-f-83
40.53
0.3
48.27
0.1
50.87
0.7
41.30
0.05
hec-s-93
15.01
0.02
19.07
0.01
15.31
0.1
15.31
0.05
kfu-s-93
18.07
1.2
18.25
0.3
21.20
3.2
17.49
0.4
lse-f-91
13.8
0.4
14.43
0.3
16.56
1.91
13.30
0.3
sta-f-83
169.71
0.1
164.95
0.01
167.28
0.05
170.35
0.02
tre-s-92
10.63
0.3
10.36
0.2
11.21
1.0
10.14
0.1
uta-s-93
4.48
2.5
4.02
1.0
4.82
11.8
3.73
1.3
ute-s-92
33.82
0.1
36.28
0.03
32.77
0.2
33.62
0.03
yor-f-83
45.95
0.3
48.02
0.2
46.68
0.5
46.61
0.06
Total
363.20
8.2
374.63
4.2
379.61
38.2
362.23
4.9
Data
Table 3. Results achieved when using different initial orderings
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
38
N = number of events
N = number of exams / 2
Average
Average
Best
Worst
Average
Average
Best
Worst
Cost
Time(s)
Cost
Cost
Cost
Time(s)
Cost
Cost
car-f-92
6.01
24.1
5.72
6.44
5.93
11.2
5.50
6.22
car-s-91
6.89
40.3
6.71
7.10
7.15
20.3
6.68
7.38
ear-f-83
49.56
1.3
48.23
51.42
50.30
0.7
47.98
53.41
hec-s-93
16.59
0.2
15.03
17.52
17.09
0.1
15.01
22.23
kfu-s-93
20.62
13.2
19.70
22.02
20.31
8.4
18.49
21.77
lse-f-91
16.10
8.3
15.24
16.73
15.81
4.5
14.89
16.68
sta-f-83
169.07
0.3
162.38
173.04
172.09
0.1
166.32
183.55
tre-s-92
11.04
2.5
10.34
11.60
11.32
1.5
11.20
11.46
uta-s-93
4.80
30.2
4.68
4.90
5.04
19.7
4.66
5.61
ute-s-92
35.22
0.7
32.16
38.26
36.25
0.3
32.39
38.30
yor-f-83
49.13
49.1
47.61
50.59
47.97
48.0
45.85
49.97
Total
385.03
170.2
367.80
399.62
389.26
114.8
368.97
416.58
Data
Table 4. Results when using a randomised ordering
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
39
Random (average)
Data
LD initial order
SmD initial order
(N = number of events)
Cost
Time(s)
Cost
Time(s)
Cost
Time(s)
car-f-92
5.12
0.2
5.39
0.7
5.30
1.8
car-s-91
5.94
0.5
6.41
1.0
6.30
2.5
ear-f-83
41.83
0.07
48.84
0.1
47.10
0.1
hec-s-93
15.03
< 0.001
13.59
0.04
15.92
0.04
kfu-s-93
19.24
0.3
20.44
0.6
19.92
0.9
lse-f-91
14.01
0.2
15.93
0.35
15.40
0.5
sta-f-83
178.29
0.01
173.58
0.01
171.08
0.04
tre-s-92
10.04
0.04
10.49
0.3
10.31
0.4
uta-s-93
3.90
1.0
4.40
1.8
4.39
1.7
ute-s-92
33.63
0.2
37.59
0.07
34.76
0.06
yor-f-83
46.72
0.4
47.56
0.12
47.67
0.2
Total
373.75
2.9
384.22
5.0
378.15
8.2
Table 5. Results when considering hard constraints and using exponent based adaption
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
40
Data
Using LD initial
Using flat initial
Using SatD Initial
Using SmD Initial
ordering
ordering
Ordering
Ordering
Cost
Time(s)
Cost
Time(s)
Cost
Time(s)
Cost
Time(s)
car-f-92
4.44
26.3
4.39
17.9
4.53
26.8
4.55
16.6
car-s-91
5.55
28.75
5.13
21.3
5.41
3.9
5.33
32.7
ear-f-83
38.09
2.74
37.63
0.6
37.42
3.7
38.36
3.8
hec-s-93
12.58
0.1
12.37
0.7
12.06
0.8
12.39
0.7
kfu-s-93
16.35
11.3
15.85
14
15.64
11.8
16.50
21.9
lse-f-91
11.71
11.3
12.08
6.6
11.72
4.3
12.45
9.4
sta-f-83
168.68
1.3
164.95
0.01
168.58
1.3
162.09
0.7
tre-s-92
8.67
4.1
8.96
4.3
8.85
5.90
8.87
3.2
uta-s-93
3.45
34.8
3.45
9.8
3.46
31.7
3.60
28.6
ute-s-92
27.41
2.5
28.07
2.4
27.9
3.0
28.52
1.5
yor-f-83
41.80
0.8
42.46
2.8
43.07
1.3
42.35
1.9
Total
338.73
124.0
335.34
80.4
338.64
94.5
335.01
121.0
Table 6. Results when considering soft constraints when incrementally adapting
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
41
Using LD initial
Using SatD Initial
Using SmD Initial
ordering
Ordering
Ordering
Data
Cost
Time(s)
Cost
Time(s)
Cost
Time(s)
car-f-92
4.41
25.6
4.37
13.4
4.68
21.7
car-s-91
5.20
42.4
5.17
8.4
5.51
34.9
ear-f-83
38.85
3.3
38.63
2.3
37.58
3.6
hec-s-93
12.82
0.1
12.61
0.2
12.20
0.4
kfu-s-93
15.35
9.9
15.99
19.8
15.88
4.0
lse-f-91
11.73
4.4
11.64
5.7
12.53
1.7
sta-f-83
165.01
1.5
167.33
1.3
161.91
0.2
tre-s-92
8.65
1.1
8.72
1.4
8.85
1.8
uta-s-93
3.43
35.6
3.40
25.3
3.56
30.6
ute-s-92
29.58
0.2
29.11
2.7
29.67
0.2
yor-f-83
42.94
1.4
42.80
0.7
42.11
2.8
Total
338.0
125.5
339.77
81.2
334.48
101.9
Table 7. Results when considering soft constraints when exponent based adapting
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
42
Average
Data
Best
Worst
Cost
Cost
Time(s)
Cost
Time(s)
car-f-92
4.36
4.35
17.6
4.37
7.5
car-s-91
5.12
5.10
30.7
5.16
25.2
ear-f-83
37.98
36.81
1.9
38.76
0.2
hec-s-93
12.23
12.05
0.6
12.35
1.1
kfu-s-93
15.45
15.44
13.0
15.47
20.2
lse-f-91
11.35
11.29
4.1
11.40
3.5
sta-f-83
170.35
170.35
0.02
170.35
0.02
tre-s-92
8.60
8.57
5.9
8.65
4.8
uta-s-93
3.43
3.43
20.4
3.44
28.2
ute-s-92
27.77
27.57
2.4
27.91
0.6
yor-f-83
40.98
40.77
2.9
41.13
2.5
Total
337.62
335.73
99.5
338.99
93.82
Table 8. Results when considering soft constraints and using saturation degree with random based
adaption
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
43
Average
Data
Best
Worst
Cost
Cost
Time(s)
Cost
Time(s)
car-f-92
4.32
4.32
95.6
4.32
95.6
car-s-91
4.99
4.97
53.2
5.03
50.4
ear-f-83
36.29
36.16
31.5
36.38
30.8
hec-s-93
11.81
11.61
10.9
11.95
9.3
kfu-s-93
15.23
15.02
115.5
15.37
66.3
lse-f-91
11.06
10.96
57.8
11.14
68.7
sta-f-83
170.35
170.35
0.02
170.35
0.02
tre-s-92
8.44
8.38
54.7
8.55
51.2
uta-s-93
3.37
3.36
180.5
3.38
332.8
ute-s-92
27.71
27.42
4.9
27.91
0.6
yor-f-83
40.98
40.77
2.9
41.13
2.5
Total
334.45
333.32
607.5
335.51
708.2
Table 9. Results when considering soft constraints and using saturation degree with random based
adaption and extended run time
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
44
Tabu Solver
Data
Carter et al.
Adaptive
Di Gaspero & Schaerf
Best Cost
Time(s)
Costs
Costs
Times (s)
car-f-92
5.2
860.6
6.2-7.6
4.32-4.68
95.6/21.7
car-s-91
6.2
30.2
7.1-7.9
4.97-5.55
53.2/28.17
ear-f-83
45.7
4.6
36.4-46.5
36.16-38.85
31.5/3.3
hec-s-93
12.4
3.7
10.8-15.9
11.61-12.82
10.9/0.1
kfu-s-93
18.0
12.3
14.0-20.8
15.02-16.50
115.5/21.9
lse-f-91
15.5
20.3
10.5-13.1
10.96-12.53
57.8/1.7
sta-f-83
160.8
3.9
161.5-165.7
161.910.2/0.02
170.53
tre-s-92
10.0
16.2
9.6-11.0
8.38-8.96
54.7/4.3
uta-s-93
4.2
50.7
3.5-4.5
3.36-3.6
180.5/28.6
ute-s-92
29.0
42.4
25.8-38.3
27.41-29.67
2.5/0.2
yor-f-83
41.0
25.2
41.7-49.9
40.77-42.94
2.9/1.4
Total
362
1010.5
327.1-381.2
324.87605.3/111.4
346.63
Table 10. Comparisons of best and worst case performance
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
45
Number
Max students per
of periods
period
car-f-92
36
2,000
kfu-s-93
21
1,955
nott
23
1,550
pur-s-93
30
5,000
Data
Table 11. Data used by Burke and Newall [Burke and Newall, 1999]
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
46
TS
Decomposition
Data
Adaptive (LD)
Adaptive (Flat)
Adaptive (SatD)
time(s)
cost
time(s)
cost
time(s)
cost
time(s)
Solver
average
average
cost
cost
car-f-92
3377
1765
186
1989
13.9
1921
8.9
1894
10.4
kfu-s-93
1845
1626
105
1728
8.1
1907
0.9
1769
7.6
nott
810
552
156
548
19.4
584
14.7
597
27.1
pur-s-93
126046
80323
1522
116660
16.0
121412
153.1
94364
45.3
Total
132078
84266
1969
120925
57.4
125824
177.6
98624
90.4
Table 12. Comparison with Burke and Newall [Burke and Newall, 1999]
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
47
Data
TS Solver
Decomposition
Adaptive
Average
Best
Worst
average
best
average
best
Cost
Cost
time(s)
Cost
time(s)
car-f-92
3377
3048
1765
1665
1777
1775
95.0
1778
40.6
kfu-s-93
1845
1733
1626
1510
1459
1422
14.4
1514
38.0
nott
810
751
552
519
555
545
157.4
562
58.4
pur-s-93
126046
123935
80323
63824
98382
97237
1650.0
99146
801.0
Total
132078
129467
84266
67518
102173
100979
1916.8
103000
938.0
Table 13. Comparison with Burke and Newall when using randomised adaption
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
48
Figures
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
49
FOR each exam e in order of decreasing estimated difficulty difficulty(e)
IF e can be schedule in the timetable THEN
schedule e in the period of least penalty
ELSE
leave e unscheduled
ENDIF
ENDFOR
Figure 1. Pseudo-code for the basic heuristic approach
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
50
FOR i := 1 TO #<max number of iterations>
FOR each exam e in order of decreasing difficulty(e, i)
IF e can be schedule in the timetable THEN
schedule e in the period of least penalty
heurmodei+1 := heurmodei
ELSE
heurmodei+1 := modify(heurmodei)
ENDIF
ENDFOR
evaluate solution and store if best found so far
ENDFOR
Figure 2. Sample pseudo-code for the adaptive method
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
51
FOR each exam e
modifiede = 0
ENDFOR
maxModify = 100
FOR i := 1 TO <max number of iterations>
wasModified := false
FOR each exam e in order of decreasing difficulty(e, i)
IF e can be schedule in the timetable AND Penalty(e) < maxCost THEN
schedule e in the period of least penalty
IF modifiede < maxModify THEN
wasModified := true
ENDIF
heurmodei+1 := heurmodei
modifiede := modifiede + 1
ELSE
heurmodei+1 := modify(heurmodei)
ENDIF
ENDFOR
evaluate solution and store if best found so far
IF NOT wasModified THEN
maxCost := worst cost encountered in last timetable
ENDIF
ENDFOR
Figure 3. Sample pseudo-code for the adaptive method when considering soft constraints
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
52
Figure 4. Incremental and Exponent adaption on random orderings for the car-f-92
problem
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
53
Figure 5. Largest degree and saturation degree when adapting on soft constraints
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
54
Figure 6. Incremental and Exponential Adaption on smallest degree orderings
for the car-f-92 problem
Solving Examination Timetabling Problems through Adaptation of Heuristic Orderings
55