High school timetable modeling in the situation of external
disturbances
M. Veenstra (s1796399)
Masterthesis EORAS
Supervisor: Prof. Dr. I.F.A. Vis
High school timetable modeling in the situation of
external disturbances
M. Veenstra (s1796399)
December 10, 2012
Abstract
1
Introduction
In high schools the generation of timetables is a time consuming process. The general timetabling problem is proven to be NP-complete [15]. Observations in practice learn that timetables are generated by the use of software and the interaction of a scheduler. External disturbances are commonly penetrated in a straightforward way. External disturbances, such as the absence of a teacher due to illness or absence of teachers and classes due to an activity week1, do in most
cases lead to schedules with more idle times2than necessary. Changes made in the timetable due to external disturbances are mainly arranged by the manual interaction of the scheduler. The manual interaction of a scheduler requires a personal effort and therefore time, moreover the quality of the new schedule is possibly less than if obtained by software based on a mathematical model.
The goal of this paper is to model high school timetables in the situation of external disturbances. Instead of generating an entirely new timetable, an already existing timetable is modified. A so-lution method is developed to generate an adjusted timetable meeting the requirement of a small overall number of idle times while keeping the timetable consistent. In this context consistency means that the new timetable does not deviate too much from the initial timetable. The deviation can be stated in terms of the number of shifted lectures and the number of time periods between the time period of a lecture in the new schedule compared to the old schedule.
Two types of high school timetabling problems can be considered, namely one where all students in a class follow exactly the same lectures and one where students in a class may attend different lectures [19]. The solution method developed in this paper is for high schools where all students in a class follow exactly the same lectures. In the lower classes in the educational system in The Netherlands all students in a class follow exactly the same lectures, which is also the case for some other countries, e.g., the 11- to 14- year-olds in English Secondary Schools. For an overview of the timetabling problem in different countries see [19]. In The Netherlands it is common prac-tice to give a higher importance for rescheduling lower classes to minimize the amount of idle times compared to higher classes. Pupils in the latter category are capable of working more in-dependently. Note that the method developed in this paper is also applicable to high schools where students in a class may attend different lectures, in this case only the lectures attended by all students in a class are allowed to be shifted.
In the literature three groups of timetabling problems can be distinguished covering the timetable problems in schools and universities. These groups are the examination, course and school timetabling problems. Examination timetabling consists of scheduling exams of a school or uni-versity, such that exams with common students do not overlap and the exams are dispersed over the periods as much as possible for the students. For a complete description and an overview of solution approaches see e.g., the papers of Schaerf [23], Qu et al. [20] and Carter & Laporte [6]. Course timetabling consists of scheduling lectures of a set of (university) courses, where the overlap of lectures of courses which have common students is to be minimized [23] (e.g., the papers of Schaerf [23] and Bonutti et al. [4]). The school timetabling problem, also referred to as the class-teacher model, consists of assigning lectures to periods such that no teacher or class is involved in more than one lecture at a time [11]. For the school timetabling problem several theoretical papers are written, see e.g. the papers of De Werra ([11], [12]) and Carter & Laporte [7]. The model given in this paper is a special case of the school timetabling problem.
1During an activity week activities are scheduled for certain classes accompanied by a set of teachers. The regular
lectures for those teachers and classes are removed.
2An idle time for a class is a time period without a scheduled lecture, such that there is a scheduled lecture earlier and
There are a large number of papers considering the general school timetabling problem, exten-sions and special cases of the problem. Examples include the addition of the assignment of teach-ers to classes [9], the incorporation of the availability of rooms and the addition of lectures given simultaneously to more than one class [26], [22]. The general form of the class-teacher model and some extensions are described in the paper of De Werra [11]. The paper of Even et al. [15] treats the complexity of the school timetabling problem and shows that all common timetable problems are NP-complete.
Different solution techniques are considered for the school timetabling problem. A reduction to the graph coloring problem is described by Neufeld and Tartar [17]. Tripathy [24] presents a large integer linear programming problem, using a solution method based on Lagrangean relaxation coupled with subgradients optimization incorporated with a branch and bound procedure. Pa-poutsis et al. [18] solve the problem using a column generation approach, by modeling the school timetabling problem as a set partitioning problem. Due to the amount of time it takes to find a solution to the problem, and by the complexity of the problem, a large number of papers are dedicated to heuristics. Among these are neural networks [5], genetic algorithms [8], tabu search [10], [22] and simulated annealing [1], [27].
In the literature, high school timetable modeling mainly has the aim to generate a new timetable. In this paper instead of starting from scratch we start with a given timetable. The given timetable is subject to external disturbances. Disturbed lectures are removed from the schedule, yielding a feasible timetable. A solution method is described which modifies the given timetable such that the number of idle times is decreased. The methods in the literature dealing with the im-provement of a given timetable, such as the paper of Zhang et al. [27] considering a simulated annealing approach, try to find better solutions by breaking out of a local minimum, yielding to a quite different timetable. Methods such as tabu search, genetic algorithms and simulated annealing are developed to have a large search space in order to find solutions breaking out of a local optimum. In the situation considered in this paper consistency in the timetables is required, since in practice timetables which deviate much from the old timetable will not be implemented. Hence it is not desirable to break out of a local optimum. Therefore the addition of the consis-tency requirement to the problem forces the need for new solution methods for the improvement of the timetable.
2
ILP model
In this paper the curriculum is assumed to be fixed for each class and the assignment of teach-ers to classes is given. In our model instead of considering the subject of a lecture, the teacher teaching the subject is investigated. A preassignment is a pair of a class and a teacher which have to meet at a certain time period. A combination of a class, teacher and time period in the old timetable to be kept in the new timetable, is defined as a preassignment. Unavailabilities of a teacher are periods a teacher is unavailable. These unavailabilities are different from the distur-bances, since they are given at the moment the initial timetable is generated. Room requirements could be added to the problem, however throughout this paper enough capacity is assumed and therefore, and for the easy of reading, room requirements are not taken into account. Whenever a disturbance takes place, the disturbed lecture is removed from the schedule. Note that through-out the paper the length of a lecture is set at one time period.
In the paper of De Werra [11] the class-teacher model with preassignments and unavailabilities is formulated. This problem is proven to be NP-complete. The problem formulated by De Werra is a feasibility problem, which means that the aim is to find a feasible solution. Our ILP model, given in this section, is described as an optimality problem. Constraints (1)-(3) of the ILP given in this section correspond to the formulation of De Werra, constraints (4)-(7) are based on the formulation of Santos et al. [21] and constraints (8)-(14) are specific constraints for our problem. For our problem we need to specify the following data:
• m: Number of classes; • n: Number of teachers; • d: Number of days;
• p: Number of time periods over d days; • Pq0: Set of time periods for day q, where
d
[
q=1
Pq0 = {1, . . . , p};
• C = {c1, . . . , cm}: Set of classes, where a class consists of a set of students who follow
exactly the same program; • T = {t1, . . . , tn}: Set of teachers;
• R = (rij): m×n requirement matrix, where rij is the number of required lectures over d
days, involving class ci and teacher tj, minus the disturbed lectures between class ci and
teacher tj;
• djk =
(
1 if teacher tjis disturbed at period k
0 otherwise
• Oijk =
(
1 if class ciand teacher tjdo meet at period k in the old schedule
• pijk =
(
1 if class ciand teacher tjmeet at period k for a lecture which is a preassignment
0 otherwise
• bik=
(
1 if class ciis available and not preassigned at period k
0 otherwise
• cjk =
(
1 if teacher tjis available and not preassigned at period k
0 otherwise
• bij: Maximum number of time periods a lecture for class cigiven by teacher tjis allowed to
be shifted to a period earlier in time, compared to the time period of the lecture in the old schedule;
• uij: Maximum number of time periods a lecture for class ci given by teacher tjis allowed
to be shifted to a period later in time, compared to the time period of the lecture in the old schedule;
• w1, w2, w3: Nonnegative weights, where w3>>w1, w2;
• rij=rij− p
∑
k=1
pijk.
Note that the time periods on a day, which is not the first day, do not start again at one but con-tinue from the day before.
The (decision) variables are given by:
• aiq: First time period with some activity in the schedule for class ciat day q;
• aiq: Last time period with some activity in the schedule for class ciat day q;
• viq=
(
1 if class cihas some activity at day q
0 otherwise
• hiq: number of idle times in schedule of class ciat day q;
• xijk=
(
1 if class ciand teacher tjmeet at period k for a lecture which is not a preassignment
0 otherwise
• xijk=xijk+pijk: determines whether class ciand teacher tjmeet at period k;
• xij0=0;
• sijk=
(
1 if Oijkis not equal to xijk
0 otherwise
• eijl: Time period of the l-th lecture, over d days, of class ciby teacher tjin the old schedule,
where l = {0, . . . , rij}and eij0=0;
• eijl: Time period of the l-th lecture, over d days, of class ciby teacher tjin the new schedule,
The ILP model is formulated by: minimize m
∑
i=1 w1 d∑
q=1 hiq+w2 n∑
j=1 p∑
k=1 sijk ! +w3 m∑
i=1 n∑
j=1 rij∑
l=0 (eijl+eijl) s.t. p∑
k=1 xijk =rij i=1, . . . , m; j=1, . . . , n (1) n∑
j=1 xijk ≤bik i=1, . . . , m; k=1, . . . , p (2) m∑
i=1 xijk ≤cjk−djk j=1, . . . , n; k=1, . . . , p (3) viq≥ n∑
j=1 xijk i=1, . . . , m; q=1, . . . , d; ∀k∈Pq0 (4) aiq≤ (q· |Pq0| +1) − (q· |Pq0| +1−k) n∑
j=1 xijk i=1, . . . , m; q=1, . . . , d; ∀k∈Pq0 (5) aiq≥k n∑
j=1 xijk i=1, . . . , m; q=1, . . . , d; ∀k∈Pq0 (6) hiq≥aiq−aiq+viq− n∑
j=1k∈q∑
d xijk i=1, . . . , m; q=1, . . . , d (7) eijl∑
k=0 Oijk =l i=1, . . . , m; j=1, . . . , n; l=0, . . . , rij (8) eijl∑
k=0 xijk =l i=1, . . . , m; j=1, . . . , n; l=0, . . . , rij (9) eijl−eijl≤bij i=1, . . . , m; j=1, . . . , n; l=0, . . . , rij (10) eijl−eijl≤uij i=1, . . . , m; j=1, . . . , n; l=0, . . . , rij (11)(1−Oijk) +xijk+sijk ≥1 i=1, . . . , m; j=1, . . . , n; k=1, . . . , p (12)
sijk+Oijk+ (1−xijk) ≥1 i=1, . . . , m; j=1, . . . , n; k=1, . . . , p (13)
xijk =xijk+pijk i=1, . . . , m; j=1, . . . , n; k=1, . . . , p (14)
xijk, xijk ∈ {0, 1} i=1, . . . , m; j=1, . . . , n; k=1, . . . , p (15)
eijl, eijl∈ {0, . . . , p} i=1, . . . , m; j=1, . . . , n; l=0, . . . , rij (16)
In our model shifts of lectures take place in order to reduce the number of idle time periods. One of the requirements of the timetable is consistency, therefore tight bounds, modeled by constraints (10) and (11), are set at the number of time periods a lecture is allowed to be shifted to a period earlier and later in time, compared to the time period of the lecture in the old schedule. Also the number of shifts determine the consistency of the timetable, the number of shifts is a performance measure and minimizing the number of shifts is part of the objective. Minimizing the number of shifts and minimizing the number of idle time periods are conflicting goals, therefore a balance between those goals has to be found.
The objective function of the ILP consists of two parts. The first part is a weighted average of the number of idle times over all classes and days and the number of shifts over all classes, teachers and days. The weights are dependent on the conceptions of each particular high school. The second part of the objective function is a modeling trick to make sure that, in combination with constraints (8) and (9), the variables eijland eijlare set in a proper way.
Constraint (1) takes care that the requirement of the number of lectures a class and teacher are involved are met for the meetings which are not a preassignment. Constraint (2) provides that a teacher can only be assigned to a class when this class is available. Since bikis a binary variable
this constraint also provides that at most one teacher can be assigned to a class at a given time period. Constraint (3) makes sure that a teacher is teaching a class only when the teacher is available, and a teacher is teaching only one class at a time. Constraint (4) indicates whether there is some activity for class ciat day q. Constraints (5), (6) and (7) together indicate hiq, the number
of idle time periods for a given class ci at a given day q. Constraint (5) determines the time
period of the first non-empty period for each day and for each class. Constraint (6) determines the time period of the last non-empty period for each day and each class. Constraint (8) and (9), in combination with the objective, set the variables eijland eijlin a proper way, namely such that
eijland eijlis set to be the time period of the l−th lecture between class ciand teacher tjover d
days in the old, respectively new, schedule. Constraint (10) and (11) take care that the maximum number of time periods a lecture for class cigiven by teacher tj is allowed to be shifted forward
and backwards in time is satisfied. Whenever a meeting between class ciand teacher tjis shifted
from time period k1in the old schedule to time period k2in the new schedule, Oijk1 6= xijk1 and
Oijk2 6= xijk2. Constraint (12) and (13) determine the number of shifts, sijkequals one if and only if
Oijk 6= xijkand thus sijk =2, which corresponds to one shift. Constraint (14) defines the variable
xijk.
2.1
Extension
In the ILP given above, whenever there is a disturbance, the disturbed lecture is not rescheduled. This ILP can also be used to reschedule disturbed lectures. In this new situation Oijk should be
replaced by Oijk. Moreover, a small adjustment has to be made in order to remove disturbed
lectures whenever no feasible solution can be found. A penalty is to be given to the number of removed lectures. This penalty can be incorporated in the objective. Using this formulation the ILP can be used in the situation whenever the school demands for rescheduling instead of removal of the disturbed lectures.
2.2
Complexity
Considering the complexity of the problem, two extreme cases are to be distinguished. Consider in the first place the model where the parameters lijand uijare set to zero for each class ciand
teacher tj. This imposes that no lecture is allowed to be shifted and the new schedule is equal to
In the second case the parameters lij and uijare set to+∞ for all classes ciand teachers tj. This
means that there are no bounds on the number of periods a lecture is allowed to be shifted. This would yield the general class-teacher timetabling problem with preassignments and availabilities (including an objective), which is proven to be NP-complete. Given the values of lijand uij the
problem can be NP-complete, due to the NP-completeness of the underlying problem. To be able to meet the requirement to quickly generate a new solution, in the next section a heuristic is developed in order to solve the problem.
3
Heuristic approach
In this section, we first describe a policy commonly used in practice. We illustrate drawbacks of this approach as a step towards designing our new heuristic. Our heuristic is based on principles of rescheduling production planning techniques. Consequently, we first discuss some relevant aspects of this category of literature before deriving our heuristic.
3.1
A commonly used policy in practice
At the moment in practice, whenever a disturbance takes place, a scheduler randomly picks one of the idle time periods caused by the disturbance, and tries to fill this period by shifting the last or first lecture of that class at that day to the idle time period. Whenever this is not possible, the scheduler tries to find another lecture of that class at that day which can be shifted to the idle time period, such that the last or first lecture of that day can be shifted to the time period of that lecture. Whenever this is not possible either, the scheduler checks whether a lecture of that class at the end or start of another day can be shifted to the idle time period. In summary,
1. Create a list with idle time periods caused by disturbances, over all classes, sorted by time. 2. If list of idle time periods still contains idle time periods: select an idle time period for a
class; If not, go to step 11.
3. Check if the last lecture of the day can be moved to the idle time period. • If yes, make move, delete idle time period from the list and go to step 2; • If no, go to step 4.
4. Check if the first lecture of the day can be moved to the idle time period. • If yes, make move, delete idle time period from the list and go to step 2; • If no, go to step 5.
5. Check if another lecture of the day can be moved to the empty period and the last lecture of the day can be moved to the time period of that lecture (allow 2-step move).
• If yes, make moves, delete idle time period from the list and go to step 2; • If no, go to step 6.
6. Check if another lecture of the day can be moved to the empty period and the first lecture of the day can be moved to the time period of that lecture (allow 2-step move).
7. Make a list of days in the same week which follow the day at which the idle time period occurs.
8. If list of days still contains days: select the first day; If not, delete idle time period from the list and go to step 2.
9. Check if the last lecture of the selected day can be moved tot he idle time period; • If yes, make move, delete idle time period from the list and go to step 2; • If no, go to step 10.
10. Check if the first lecture of the selected day can be moved to the idle time period; • If yes, make move, delete idle time period from the list and go to step 2; • If no, delete day from the list and go to step 8.
11. End
Note that in our experiments the procedure is modeled in such a way that the idle time periods due to disturbances are processed in sequence of time. The pseudo code of this policy can be found in Figures 1-4.
We note the following drawbacks of this approach. Whenever an idle time period is investi-gated, the scheduler does not reconsider this idle time period at a later stage again. Secondly, the scheduler only considers the idle time periods caused by disturbances and not the ones that were already part of the original schedule. Next to that, the sequence of idle time periods to be filled are chosen randomly, without a priority rule.
Our heuristic will prioritize the idle time periods and the lectures to be shifted. Moreover, when an idle time period is not filled, there is a probability that this time period is considered again in another replication. The heuristic takes into account all idle time periods in the schedule of a class which is affected by a disturbance. Due to the large number of runs, different sequences of idle time periods and lectures to be shifted are investigated. This might give a higher probability of finding a better final schedule compared to the policy commonly used in practice. We will perform several experiments to test this.
3.2
Heuristic approach
The heuristic developed in this section is based on ideas and characteristics developed for pro-duction (re)scheduling. Many papers can be found for propro-duction scheduling in different settings and a lot of attention is paid to production rescheduling. Finding an optimal production sched-ule in e.g. a job shop setting is known to be NP-hard [3]. A short literature review shows that mainly heuristics are developed for these problems.
processes in the timetabling problem in high schools, moreover, the number of disturbances is higher in most cases.
The heuristic developed in this section is based on a combination of an insertion technique and priority ruling. An example of an insertion technique used for project scheduling is given in the paper of Artigues et al [2]. The developed heuristic also makes use of priority rule based schedul-ing which is an important heuristic solution technique in production schedulschedul-ing. A description of priority rule based scheduling can be found in the paper of Kolisch [16].
Parameterized regret-based random sampling is one of the methods of priority rule based schedul-ing. It is introduced by Drexl [13] and Drexl and Grünewald [14]. It accounts for the relative difference in priorities and a regret factor is given for not choosing a given element. Given the priority for each element i of a set D, the regret factor is given by: wj =vj−min
i∈Dvi. The
proba-bility for choosing an element equals pj= (wj+1)α
∑
i∈D (wi+1)α .An α equal to zero corresponds to random sampling, where an α approaching infinity corre-sponds to deterministic selection.
Parameterized regret-based random sampling is used in our heuristic to account for the relative difference in priorities instead of only the difference in priorities. Consider as an example the case that probabilities are to be given to different periods. The probability for period i is defined by its priority, which can for example be defined by the number of time periods between period i and k. A list of periods i can be set up such that the periods are sorted on priority. The proba-bility of a period can then be determined by the position on this list, hence the relative difference between the periods are not accounted for in the probabilities. Instead of using that approach we use parameterized-regret based random sampling such that the number of time periods between the periods is accounted for in the probability given to each period.
The pseudo code of our heuristic for the rescheduling of high school timetables can be found in Figure 5, a description of the heuristic is given in this subsection.
Each time a set of disturbances is given, a given number of runs is carried out independently, where the best solution over these runs is the output of the heuristic.
At the start the disturbed lectures are removed from the schedules, the combination of the sched-ules of the classes which are affected by the disturbances is stored and called the initial schedule. The initial schedule and its objective value are stored as the best solution so far for each run. The objective value equals a weighted average of the number of shifts and the number of idle periods. The availability of the teachers is adapted to the disturbances. By choosing the initial schedule as the combination of all schedules, instead of the combination of schedules of only those classes which are affected by the disturbances, the schedules of the classes which are not affected by the disturbances are also allowed to change. However, due to practical reasons, in practice the schedules of the classes which are not disturbed are commonly not taken into account.
For each run a number of replications is carried out. Each replication continues on the schedule found in the replication before. Whenever a replication yields a better objective value than the best objective value found so far for this run, the best solution for this run is changed to this new solution.
In each iteration an empty period3corresponding to a class is sampled, as long as there are still
time period is equal. Whenever an empty period which is not an idle time period is sampled, the probability of picking an empty period which is based on parameterized regret-based ran-dom sampling and is dependent on the distance between the period and the smallest distance between this period and a lecture on that day. The probability decreases when this distance in-creases. Whenever the empty period corresponding to a class is sampled, a lecture of this class is sampled. The lecture is sampled from a sample of lectures corresponding to a teacher which is available at the empty time period. The priority of a lecture to be chosen is higher when it is on the same day as the empty period and whenever it is the last or first lecture of that class on a day. The priorities determine probabilities based on parameterized regret-based random sampling. Whenever the sample of lecture where the corresponding is available at the empty time period is empty, the iteration is ended, otherwise the lecture is shifted and the availability of the teacher is adjusted. In summary,
1. Create a list of classes affected by the disturbances, set the schedule for these classes by deleting the disturbed lectures. Set the schedule as the best solution so far. Set the value of Run and Replication equal to zero.
2. Check if Run is smaller than NrRuns; • If yes, Run=Run+1 and go to step 3; • If no, go to step 9.
3. Check if Replication is smaller than NrRepl;
• If yes, Replication=Replication+1 and go to step 4; • If no, go to step 2.
4. Create a list of idle time periods over all classes in the list of classes. 5. Check if the list of idle time periods is empty.
• If yes, go to step 2;
• If no, sample a value between zero and one and check whether the value is smaller than w3;
– If yes, randomly pick one of the idle time periods, call it Insert;
– If no, create a list of empty time periods which are no idle time periods over all classes in the list of classes;
∗ Check if list of empty time periods which are no idle time periods is empty; · If yes, randomly pick one of the idle time periods, call it Insert;
· If no, pick one of the elements in the list using parameterized-regret based random sampling, where the probabilities are determined by the distance between the period and the first or last lecture of the day, call it Insert. 6. Create a list of lectures of the class corresponding to Insert, such that the teacher
corre-sponding to the lecture is available at time period Insert. 7. Check if the list of lectures is empty;
• If yes, go to step 3;
Insert and whether it is the first or last lecture of the day, call it Delete. Make move of lecture Delete to time period Insert and go to step 8. 8. Check if the new schedule is better than the schedule found so far;
• If yes, set the schedule as the best solution so far and go to step 3; • If no, go to step 3.
9. Set the schedule to the best schedule over all runs. 10. End
3.3
Extension
The heuristic can also be extended to the situation where a disturbed lecture is rescheduled. Then the disturbed lectures should also be considered in the set of non-empty periods to be deleted, in order to be inserted in the schedule at an empty time period. A penalty is to be given to the number of disturbed lectures which cannot be inserted. The priorities should be altered to this new setting.
4
Experiments
In this section we perform both a numerical and practical validation. First, several experiments are performed to set the values of the parameters as defined in the heuristic. Secondly, we test the quality of the heuristic by comparing the output with the solutions provided by the ILP for a set of small instances. Finally, we compare the performance of our heuristic with the policy as described in section 3.1.
For the experiments we use data obtained in practice. The dataset describes the schedules of the lower classes and the availability of the teachers. The data can be found in Tables 12-15. As indicated earlier, we study the lower classes. The higher classes are set as a preassignment. The number of teachers is equal to 78 and the number of lower classes equals 28. A week consists of five days with each nine time periods at which lectures are allowed to be scheduled. We consider one set of schedules for our numerical and practical validation.
We consider two kinds of commonly encountered disturbances. In the first case we define ’small’ sets of disturbances. Each of these sets contains one to six teachers, who are lecturing at Monday in the normal situation and are in this situation not available at Monday. The probability that the cardinality of a set equals one, four, five or six equals a half, the probability that the cardinality of a set is equal to two or three is also a half. This situation corresponds to a normal situation in practice where a couple of teachers are absent due to e.g. illness. The reason for choosing Monday as the day where the disturbances take place is the flexibility of shifting lectures of all days in that week. The second case contains ’large’ sets of disturbances. Each set contains ten teachers which are not available that week, which would correspond to an activity week. Note, however, that we do not consider the absence of classes.
In our experiments the disturbances are removed from the schedules and the schedule of one week is considered.
initial schedule idle time periods might exist, that could be eliminated as well by applying our heuristic. The policy in 3.1, however, only considers newly created idle time periods due to disturbances. The other solution methods also take into account the idle times already existing in the initial schedule. To avoid a bias in the comparison of the solution methods we consider all idle time periods in all three methods.
4.1
Parameters of the heuristic
In this section several experiments are carried out in order to find the values for the different parameters in the heuristic. In the first place the parameters are set for the situation where we consider large sets of disturbances. On the basis of these results the parameters for the small sets of disturbances are considered. The parameters for the larger sets of disturbances are in-vestigated first, since the number of changes to be made are probably higher for these sets and therefore we expect these results to be more dependent to the setting of the parameters compared to the results for the smaller sets.
In Table 1 the settings of the experiments can be found. The results of these experiments can be found in Table 2-6. The values in the tables correspond to the mean objective value, of the sum over all schedules of the disturbed classes over one week, over the disturbance sets.
When picking an empty time period, w3is the probability of choosing an idle time period. For
some experiments we try whether better results are obtained when choosing the value of w3
different for the first two/third number of iterations compared to the last number of iterations. Since better solutions are found for increasing w3, in later experiments we set w3 at the same
value for each iteration. The number of disturbance runs is the number of independent sets of disturbances for which the heuristic is carried out.
The first experiment considers different values for α1 and α2, which are the parameters
deter-mining the randomness of the procedure of picking an empty period and a lecture based on parameterized regret-based random sampling. The values for these parameters seems to point to the same direction for the different choices of w1, as can be seen in Table 2, therefore we decide to
fix this value at five. Note that the choice of w1is dependent on the conception of each particular
high school and determines the relationship between the penalty for an idle time period and the shift of a lecture. The best value found for α1 is 10000, note however that in later experiments
w3is set to one and therefore the procedure where α1is addressed is never carried out. The best
found value for α2 is five in this experiment, note that this value is to be investigated in later
experiments.
In the second experiment the number of iterations is set to 750, in later experiments it is found that with the chosen parameters the number of iteration never exceeds 250 for its best solution, therefore in later experiments the number of iterations is set to 250. Since no penalty is given to the number of time periods a lecture is shifted, we decided to set wsameday, which is a value
added to the priority a lecture one the same day as the empty period, equal to zero, note that this can be changed whenever requested by the high school. Different values for w3and α2are
investigated. The results indicate that a larger w3yields better results, therefore we set w3at one
for the moment and the value of α2will be investigated in the next experiment.
The third experiment, which considers the value of α2yields a value equal to 100 for α2. Using
this value, in the fourth experiment the value of w3 is investigated. The best solution for w3
equals one, which corresponds to a solution method which only investigates idle time periods and does not tries to fill the empty periods before the time period of the first lecture or after the time period of the last lecture. Since w3equals one, we do not have to take into account α1
The different values found for the large sets of disturbances are also used for the small sets of disturbances. The value for α2is investigated for these sets of disturbances in experiment five,
a value equal to 500 is somewhat better than a value equal to 100. The differences are rather small, therefore we have chosen to set the value equal to the value found for the large sets of disturbances.
Due to the small amount of time it takes to run the heuristic and since a larger number of runs gives a higher probability for a better schedule, the number of runs for the heuristic is increased to 4000. An overview of the values for the parameters for the heuristic is given in Table 7. In practice the value for w1 can be set by the scheduler based on his view on a correct balance
between the number of shifts and the number of idle time periods. The value for wf lhcan also be
altered by the scheduler whenever he prefers shifting lectures over one day instead of over more days.
4.2
Comparison of the solution methods
For the comparison of the solution methods, the number of periods a lecture is allowed to be shifted is set to+∞, in order to have a better comparison due to the fact that the number of time
periods over which a lecture is shifted is not reflected in the objective.
For the comparison of the solution methods 1000 large sets of disturbances and 1000 small sets of disturbances are generated in the same way as described before.
First we test the quality of the heuristic. For 200 small sets of disturbances a comparison is made between the heuristic and the ILP. An overview of the mean values over the disturbance sets of the heuristic and the procedure of a scheduler is given in Table 8. The values reflect the mean value, over all disturbed classes over one week, over the disturbance sets. The heuristic uses on average eight percent more shifts, in order to eliminate 1.18 percent less idle time periods. Hence, we can conclude that there is a small gap between the heuristic and the ILP for the small instances.
Next, we perform a practical validation by comparing the performance of our heuristic with the policy described in section 3.1. The computing time for the heuristic is approximately 2 seconds per disturbance run for the small sets of disturbances. For the large sets of disturbances the com-puting time for the heuristic is approximately 25 seconds per disturbance run.
An overview of the mean values over the disturbance sets of the heuristic and the procedure of a scheduler is given in Table 9 and Table 10. Moreover, for the small sets of disturbances the mean values over disturbance sets of the same cardinality, with respect to the number of absent teachers, are given. For the large sets of disturbances, the heuristic always finds a better solution. For the small sets of disturbances the heuristic and the procedure of the scheduler find 84 times the same objective value out of 1000 disturbance runs. The heuristic finds a better solution in the other cases. For the small sets of disturbances, on average, 16.7 percent less shifts are need to eliminate 10.6 percent more idle time periods in comparison to the procedure of the scheduler. For the large sets of disturbances, using, on average, 1.3 percent less shifts, 21.7 percent more idle time periods are eliminated by the heuristic compared to the procedure of the scheduler.
Besides the reduced number of idle time periods, the number of shifts is reduced, when compar-ing the heuristic to the procedure of the scheduler, which is a measure for consistency.
5
Conclusion
This paper considered high school timetable modeling in the situation of external disturbances. The problem is a special case of the school timetabling problem. An ILP model is defined for the problem, as an extension of the formulation of De Werra [11] and Santos et al. [21]. A heuristic is developed for the problem, inspired by ideas and characteristics of production (re)scheduling heuristics.
The heuristic is compared to the ILP for small instance sets where we found that the heuristic uses on average 8 percent more shifts, in order to eliminate 1.18 percent less idle time periods. Hence, we can conclude that there is a small gap between the heuristic and the ILP for the small instances.
The heuristic is also compared to the procedure applied in practice. For the small sets of distur-bances, on average 16.7 percent less shifts are need to eliminate 10.6 percent more idle time peri-ods compared to the procedure of the scheduler. For the large sets of disturbances, on average, using 1.3 percent less shifts, 21.7 percent more idle time periods are eliminated by the heuristic compared to the procedure of the scheduler. Using the heuristic, not only the number of idle time periods is declined, but also the number of shifts. Comparing the heuristic with the procedure of the scheduler, we see a reduction of the number of idle time periods for the heuristic and the fulfillment of the requirement for consistency. We can conclude that the heuristic performs better than the procedure of the scheduler.
Since the computing time of the heuristic is low and the developed heuristic generates better so-lutions than the procedure commonly applied in practice, it is recommended to use our heuristic in practice.
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Appendix
Objective(X) =w1∑
i∈DC d∑
q=1 hiq+w2∑
i∈DC n∑
j=1 p∑
k=1 sijk DC=∅; for i=1 to m doif ∃j∈ {1, . . . , n}, k∈ {1, . . . , p}s.t. Oijk =1 and djk =1 then
DC=DCS {i} end end Xijk =Oijk, i∈DC, j∈ {1, . . . , n}, k∈ {1, . . . , p} ajk =cjk−djk, j∈ {1, . . . , n}, k∈ {1, . . . , p} for k=1 to p do for j∗∈ {1, . . . , n}do if dj∗k=1 then if∃is.t.Xij∗k=1 then Class=i s.t. Xij∗k=1;
q is the day corresponding to k; if aClass,q<k<aClass,qthen
f ound=0; ShiftLastFirstLectureSameDay; if f ound=0 then ShiftLectureSameDay; if f ound=0 then ShiftLastFirstLectureOtherDay; end end end end end end end Output: X;
TeacherLH=j, s.t. XClass,j,aClass,q =1; if aTeacherLH,k =1 then XClass,TeacherLH,k =1; XClass,TeacherLH,aClass,q =0; aTeacherLH,aClass,q =1; aTeacherLH,k=0; f ound=1; end if f ound=0 then
TeacherFH=j,s.t.XClass,j,aClass,q =1;
if aTeacherFH,k =1 then XClass,TeacherFH,k =1; XClass,TeacherFH,aClass,q =0; aTeacherFH,aClass,q =1; aTeacherFH,k=0; f ound=1; end end
for r∈Pq0do if found=0 then if n
∑
j=1 XClass,j,r =1 thenTeacherLH=j, s.t. XClass,j,aClass,q =1;
TeacherFH=j, s.t. XClass,j,aClass,q =1;
Teacher=j, s.t. XClass,j,r=1; if aTeacher,k=1 then if aTeacherLH,r=1 then XClass,Teacher,k =1; XClass,Teacher,r=0; XClass,TeacherLH,r =1; XClass,TeacherLH,aClass,q =0; aTeacher,r=1; aTeacher,k=0; aTeacherLH,aClass,q =1; aTeacherLH,r=0; f ound=1; else if aTeacherFH,r=1 then XClass,Teacher,k =1; XClass,Teacher,r=0; XClass,TeacherFH,r =1; XClass,TeacherFH,aClass,q =0; aTeacher,r=1; aTeacher,k=0; aTeacherFH,aClass,q =1; aTeacherFH,r=0; f ound=1; end end end end end end
if q<d then
for day=q+1 to d do if f ound=0 then
TeacherLH=j,s.t. XClass,j,aClass,day=1;
if aTeacherLH,k =1 then XClass,TeacherLH,k =1; XClass,TeacherLH,aClass,day =0; aTeacherLH,aClass,day =1; aTeacherLH,k=0; f ound=1; end if f ound=0 then
TeacherFH=j, s.t. XClass,j,aClass,day =1;
if aTeacherFH,k =1 then XClass,TeacherFH,k =1; XClass,TeacherFH,aClass,day =0; aTeacherFH,aClass,day =1; aTeacherFH,k=0; f ound=1; end end end end end
Objective(X) =w1
∑
i∈DC d∑
q=1 hiq+w2∑
i∈DC n∑
j=1 p∑
k=1 sijkSet values of w1, w3, wsameday, wf lh, α1, α2, NrRuns, NrRepl;
DC=∅, Run=0, Repl=0; for i=1 to m do
if ∃j∈ {1, . . . , n}, k∈ {1, . . . , p}s.t. Oijk =1 and djk =1 then
DC=DCS{ i} end end for i∈DC do (Xijk)0=Oijk, j∈ {1, . . . , n}, k∈ {1, . . . , p} end (ajk)0=cjk−djk, j∈ {1, . . . , n}, k∈ {1, . . . , p}
while Run<NrRuns do Run=Run+1;
Xijk = (Xijk)0, i∈ DC, j∈ {1, . . . , n}, k∈ {1, . . . , p};
ajk = (ajk)0, j∈ {1, . . . , n}, k∈ {1, . . . , p};
BestRun=X;
while Repl<NrRepl do Repl=Repl+1; for i∈DC do Idlei=∅, Emptyi=∅; for k=1 to p do if n
∑
j=1 Xijk =0 then if aiq<k<aiqthen Idlei= Idlei∪ {k} else Emptyi=Emptyi∪ {k} end end end end if [ i∈DC Idlei6=∅ then R=rand[0, 1]; InsertDeleteAdjust end end endX∗=Bestgs.t. Objective(Bestg) ≤Objective(Besth), h∈ {0, . . . ,(NrRun−1)};
Output: X∗;
if R<w3then
Randomly choose Class=i and Insert=k s.t. i∈DC, k∈ Idlei
else
if|Empty| =∅ then
Randomly choose Class=i and Insert=k s.t. i∈DC, k∈ Idlei
else for r=1 to|Empty|do l=Emptyr if l>aiqthen uEmptyr =l−aiq else uEmptyr =aiq−l end end for r=1 to|Empty|do vEmptyr = max
l∈{1,...,|Empty|}uEmptyl−uEmptyr
end
wEmptyr =vEmptyr− min
l∈{1,...,|Empty|}vEmptyl; pEmptyr = (wEmptyr+1)α1
∑
l∈{1,...,|Empty|} (wEmptyl+1) α1, r∈ {1, . . . ,|Empty|};Insert=Random element from Empty based on pEmptyr;
Class=class corresponding to Insert; end end Lectures=∅; for k=1 to p do if n
∑
j=1 XClass,j,k =1 thenif ajk =1 for j s.t. XClass,j,k =1 then
Lectures= LecturesS{ k} end end end if Lectures6=∅ then for r=1 to|Lectures|do
vLecturesr =wsameday∗1sameday+wf lh∗1f lh
end
wLecturesr =vLecturesr− min
l∈{1,...,|Lectures|}vLecturesl; pLecturesr = (wLecturesr+1)α2
∑
l∈{1,...,|Lectures|} (wLecturesl+1) α2, r∈ {1, . . . ,|Lectures|};Delete=Random element from Lectures based on pLecturesr;
Teacher=j s.t. XClass,j,Delete=1;
XClass,Teacher,Delete =0; XClass,Teacher,Insert=1; aTeacher,Delete=1; aTeacher,Insert=0;
if Objective(X) <Objective(BestRun)then
BestRun=X
end end
Experiment 1 Experiment 2 Experiment 3
NrRuns 1000 1000 1000
NrRepl 500 750 750
w1 4,6 5 5
w3for first 2/3 nr of iterations 0.9 0.97, 0.99 1
w3for rest 0.99 1,1 1
wsameday 1 0 0
wf lh 2 2 2
α1 0,1,5,10000 10000 10000
α2 0,1,5,10000 3,5,10 5,10,50,100,500,1000,5000,10000
Number of disturbance runs 391 1603 1000
Experiment 4 Experiment 5 (Small)
NrRuns 1000 1000
NrRepl 250 250
w1 5 5
w3for first 2/3 nr of iterations 0.6,0.8,0.9,0.95,0.975,1 1
w3for rest 0.6,0.8,0.9,0.95,0.975,1 1
wsameday 0 0
wf lh 2 2
α1 10000 10000
α2 100 5,10,50,100,500,1000,5000,10000
Number of disturbance runs 1748 1000
Table 1: Setting of the experiments
w1=4 w1=6 α1=0, α2=0 257.798 259.849 α1=0, α2=1 204.537 206.488 α1=0, α2=5 168.2685 170.343 α1=0, α2=10000 211.611 221.235 α1=1, α2=0 254.483 256.312 α1=1, α2=1 202.655 204.611 α1=1, α2=5 167.394 169.407 α1=1, α2=10000 210.905 220.005 α1=5, α2=0 247.790 248.857 α1=5, α2=1 199.473 200.772 α1=5, α2=5 166.506 168.419 α1=5, α2=10000 208.647 217.752 α1=10000, α2=0 221.982 223.527 α1=10000, α2=1 187.951 189.235 α1=10000, α2=5 161.465 162.811 α1=10000, α2=10000 195.708 204.281
w3=0.97, 1, α2=3 164.624 w3=0.97, 1, α2=5 161.579 w3=0.97, 1, α2=10 161.361 w3=0.99, 1, α2=3 162.607 w3=0.99, 1, α2=5 159.969 w3=0.99, 1, α2=10 159.608
Table 3: Results of Experiment 2
α1=5 159.023 α1=10 158.610 α1=50 158.653 α1=100 158.420 α1=500 158.496 α1=1000 197.396 α1=5000 197.419 α1=10000 197.507
Table 4: Results of Experiment 3
w3=0.6 166.068 w3=0.8 164.003 w3=0.9 162.450 w3=0.95 161.224 w3=0.975 160.223 w3=1 158.654
Table 5: Results of Experiment 4
α1=5 18.889 α1=10 18.965 α1=50 18.980 α1=100 18.964 α1=500 18.956 α1=1000 26.697 α1=5000 26.701 α1=10000 26.713
Table 6: Result of Experiment 5
NrRuns 1000 NrRepl 250 w1 5 w3 1 wsameday 0 wf lh 2 α2 100
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
ILP 18.192 9.096 0 10.813
Heuristic 20.278 9.823 0.126 10.687
Table 8: Mean value, over all disturbed classes over one week, over the small disturbance sets of the results of the ILP and the Heuristic
OVERALL
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 20.005 9.642 0.143 10.492
Scheduler 28.226 11.248 1.146 9.489
ONE TEACHER ABSENT
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 6.792 3.296 0.040 3.440
Scheduler 9.416 3.768 0.376 3.104
TWO TEACHERS ABSENT
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 13.180 6.404 0.075 6.965
Scheduler 18.914 7.604 0.741 6.298
THREE TEACHERS ABSENT
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 18.816 9.028 0.152 9.644
Scheduler 26.596 10.608 1.076 8.720
FOUR TEACHERS ABSENT
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 24.691 11.845 0.200 13.091
Scheduler 35.291 14.055 1.436 11.855
FIVE TEACHERS ABSENT
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 30.118 14.556 0.201 15.979
Scheduler 42.014 16.771 1.694 14.486
SIX TEACHERS ABSENT
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 34.810 16.756 0.259 18.397
Scheduler 48.664 19.181 2.060 16.595
Objective Value Nr of shifts Nr of idle periods left Nr of idle periods eliminated
Heuristic 162.096 78.598 0.98 91.111
Scheduler 245.487 79.626 17.247 74.844
Table 10: Mean value, over all disturbed classes over one week, over the large disturbance sets of the results of the heuristic and the procedure of the scheduler
Nr of idle periods eliminated
1 2 3 4 5 6 Nr of shifts = 1 2798 287 92 5 0 0 Nr of shifts = 2 488 1605 248 98 15 0 Nr of shifts = 3 11 143 202 95 2 1 Nr of shifts = 4 1 5 15 12 2 2 Nr of shifts = 5 0 1 0 1 0 0
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 2 21 12 - 10 - 19 - - - 19 2 - 10 21 12 - - - -2 6 - 25 - - - 15 6 - 17 6 6 26 6 - - - 17 - 28 3 - - - 15 14 - -4 - - - 9 - - - -5 27 - 15 26 - 7 - - - 5 26 17 - - 27 - 16 15 - 7 16 17 -6 - 25 26 27 28 - - - - 11 - - 10 - 12 - - - 8 - - - -7 26 27 23 - - 22 - - - 22 23 - - - 16 - - - - -8 - - - 11 11 16 16 - 26 26 20 20 - 14 14 9 - - - 7 9 9 18 - - 1 1 - 9 9 9 - 3 10 - - - 24 25 - - - - 12 - - 22 - - -11 24 - 27 14 - 26 6 - - - 15 - - - 17 23 - 15 -12 - - - 12 25 17 7 21 21 16 - 7 - - - 25 - 12 -13 - - - 12 - 23 3 -14 - 20 10 19 - 3 3 11 11 - - - 10 19 11 - 10 10 - - -15 - - - -16 - - - 20 - 13 - - - -17 - - - 6 - 5 4 4 - 18 18 1 - 4 18 - - 17 24 25 - - - 14 - - - - 24 - - -19 - - - 10 19 8 8 - - - 19 10 10 20 - - - - 12 - - - - 12 1 - - - 1 - - - 1 1 21 3 - - - 1 - - - 8 - - 8 - 1 - - - -22 15 15 - 17 17 - - - - 16 16 - 14 14 - - 10 - - - -23 - - - 18 - - - 19 - 21 - - - - 20 -24 - - - - 5 - - - -25 - - - 20 - - - 19 - - - -26 - 24 11 - - - 18 13 - - 25 - 13 - - 18 24 - - - -27 - - - 22 22 15 - - - 19 - 24 -28 - - - 20 12 - - - 20 - -29 - 22 20 8 - 11 12 - - - 12 - 3 - - - 19 8 - - 21 30 - - - -31 - 1 1 23 18 1 - 12 12 - - 19 1 1 23 21 18 - - 12 12 - -32 - 23 16 - - - 6 - - - 16 - - - 16 33 - - - 1 1 - - 2 - - - 21 1 - - -34 - - - - 24 - - - 5 - 24 - - - 24 - 5 35 - 13 5 21 6 16 1 - - - 22 - - - 14 2 4 -36 - - 13 9 20 - - - 13 13 7 4 13 - 12 37 - - - 18 - - - 20 19 12 - - - - 22 - 18 13 13 38 - 4 22 - - - - 20 - - - 22 - 4 - - 20 - - - -39 - - 24 - 26 - - - -40 - - - - 8 - 10 - - 9 - 8 - - 8 11 - - 23 11 10 8 8 41 9 28 - 6 - 23 14 - - - - 14 23 - - - 6 28 - - 9 42 - - 19 22 - - - 19 - - - - 19 22 43 - - - 16 6 6 11 15 44 - - - 27 28 - - 25 27 28 - 26 26 25 25 - - 21 - 27 -45 - 3 3 - 22 2 2 - - 2 2 26 3 21 28 - 23 - 3 - 21 - 20 46 - - - 4 4 20 20 - - 4 4 - - - 22 22 - - - -47 - - - -48 - 17 8 - 23 21 22 - - 3 - - - 11 - - - -49 - - - 10 - - - 23 22 - - - - 22 19 50 - 5 28 7 7 - 9 - - 14 14 7 27 5 6 28 - - - 26 5 9 -51 - - - 13 - - - 4 20 4 3 3 - - 2 - 21 11 52 - - - 3 3 - - 9 9 - - - 9 9 - 2 2 - - - - 23 23 53 - - - 11 1 8 2 18 - - - -54 - - - 9 9 - - 5 5 - 6 6 55 - 7 - 16 16 - 7 7 - - - - 16 - - 15 15 - - - 15 26 26 56 - - - 19 - - - 18 - - - 18 -57 - - - 17 10 2 - 7 - - - 7 2 58 - - - 21 - - - -59 - - - -60 - - - 17 17 - - - -61 - - 4 - - - 27 25 - - - 27 - - - 13 17 - - 17 27 - 17 62 - 10 - - 9 9 4 3 - - 13 4 - 3 - 10 - - - -63 - - - 2 2 5 5 8 8 8 8 21 21 24 24 - - - -64 - - - -65 - - - 12 - - 13 - - - -66 5 - 7 5 14 14 - - - 5 5 - - - -67 - - 21 18 - 17 17 - - - 18 25 17 - - - -68 - - - 11 11 - - - 11 - - - -69 - - - 7 24 7 - - - 28 - - 27 7 25 7 70 14 - - 25 15 - - - 28 23 - - 17 27 14 - - - 22 16 25 71 - 9 9 20 13 4 - - 3 - 3 3 - 13 20 20 - - 4 3 3 - -72 - 12 - - 21 - 11 - - - 11 - 18 73 - 6 6 15 - 6 - - - 25 25 2 24 - - - - 26 28 24 74 - - - - 27 25 - - - 26 - - - 27 - 14 - - - 16 28 - 27 75 - 2 2 13 - - - 4 2 - - 13 - - - 13 13 4 2 -76 - 14 14 - - 15 16 16 - - - 5 -77 - 26 - 28 - 24 - - - 28 - - 15 15 - 26 - - - -78 - - - 8 - - - - 8 -
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 - - - 21 10 - 19 - - - 12 - - 21 2 - 10 2 - - - 28 24 - - 26 - 7 - - 17 - - - -3 6 5 27 - 27 6 - - 5 12 - 5 - 6 - - 5 27 15 14 - -4 - - - -5 - - - 16 7 - 15 - - - 27 15 26 - 17 - - - -6 - - - 14 - - - - 9 8 - -7 - - - 22 23 - - - 22 23 - 27 - -8 17 17 7 7 - - - 27 27 - - 18 18 - - - 7 7 15 15 9 18 - 1 - - 3 3 9 - 1 1 - - - -10 - - - 25 - - - - 24 24 - - 25 - - -11 28 25 14 5 4 16 14 - - 7 24 - - - -12 16 22 22 22 23 23 17 17 7 - 23 25 - - - -13 4 18 - - 18 - 12 3 4 23 3 - - 3 18 23 23 - 12 - 4 -14 - - - 10 10 12 12 - 2 2 - - 20 - 1 1 10 10 - -15 - - - -16 - - - - 20 - - - - 13 - - - 13 20 - - - -17 3 7 - - - 2 8 8 - - - -18 - - 25 - 2 - 15 - 17 16 - - - 16 - 14 - - - -19 - - - 19 8 - - - -20 - - - 10 10 - - - - 12 12 - - - -21 - - - - 3 8 - - - 3 - - - 1 - - 3 8 - - - -22 - - - -23 - - - 19 - 21 18 - 20 - - - -24 - - 5 - - - 5 - - - -25 - 19 - - - 20 - - - - 19 - - -26 - - - 18 - 11 13 - 25 - - - 25 - 11 11 24 - - -27 - 23 23 23 21 21 24 - - 8 21 6 - - - - 24 15 - - - -28 12 - - 1 - - - 12 - 20 - - - - 1 29 11 - - 12 - - - 8 - 21 - 22 11 20 - -30 - - - - 1 2 7 - 1 6 5 3 - - - -31 - - - 12 22 1 23 - 18 - - - 22 22 21 21 1 19 19 -32 - - - 23 - - - 16 - -33 - - 2 - - - 2 2 - 1 - -34 - 14 15 - 14 5 - - - 16 14 5 - 24 - - - -35 23 15 - - - - 19 - - 20 - 21 - 18 17 3 - 7 - - - -36 - 13 - - 16 - 1 - - - - 7 - - - -37 - - - - 12 - 20 18 - - 19 - - - - 19 - - - -38 - - - 4 - 22 - 20 2 - - - 22 -39 - - - 26 26 26 24 24 - - - -40 - 9 17 - - - 8 8 11 - - - 9 8 - 10 - - - -41 5 28 16 - - 28 23 5 14 - - - 6 17 23 - 9 - 5 6 - -42 22 - - - 22 - - - - 1 - - - 9 43 1 1 - - - 15 11 - 16 9 6 1 - 15 - 1 15 16 6 11 - -44 25 26 28 - - - 25 26 27 28 - - - - -45 20 2 24 - - - - 23 22 - - - 28 20 - 26 3 24 - -46 10 10 - - - 13 13 13 - - - -47 - - - -48 - - - 5 - 10 18 11 11 - - 19 - 6 19 2 - - -49 - 20 - - 10 - - - 19 - 23 10 - - - 20 - - -50 - - - - 28 9 27 14 26 14 - - - - 6 6 26 5 27 5 - -51 19 - 9 - - - 10 1 18 8 3 22 23 12 4 52 - 3 3 - - - 6 6 - - 28 28 28 - - - 25 25 28 - - -53 2 11 - - 8 1 18 - - 10 - - - - 11 2 - - 8 18 1 -54 8 8 13 13 - - - 7 7 - - - 4 4 - -55 7 27 - - 15 27 - 16 - 11 - - - -56 - - - 19 - - - - -57 9 - 10 - 17 - 2 2 9 - 7 - - - - 7 10 9 17 - - -58 - - - 21 - - - 21 - -59 - - - -60 - - - -61 13 4 4 - - 4 4 25 - 17 - - - -62 - - - 3 4 - - 13 9 10 -63 - - - 19 19 - - - - 2 2 - - - -64 - - - 9 9 - - - -65 - 12 20 - - 20 13 20 20 - 4 - - 12 - - 4 13 - - - -66 - - - 14 - - 6 - - - 14 - 5 -67 - - - 21 - - - 17 - - 25 25 -68 - - - 11 - - - 11 - - - -69 24 16 - - - 7 - - 25 - - - -70 15 - - - - 17 25 28 28 27 - - - - 27 15 16 - 23 22 - -71 - - - 13 9 - 3 4 - 4 4 - - - -72 21 - 12 - - - 21 21 - 18 18 18 12 11 11 73 27 24 - - - 28 - - - 27 - 26 - 6 -74 26 - - - 26 16 28 14 - 16 - - -75 - - - 13 2 - - - 4 - - - -76 14 - - - 15 - - - 14 7 6 - - - -77 - - - - 24 24 - - 15 26 - - - 28 - 15 - -78 - - - 8 - - -
