Adaptive Randomized Descent Algorithm using Round Robin for Solving Course

Timetabling Problems

Anmar Abuhamdah

DMO/CAIT, Computer Science Department

Universiti Kebangsaan Malaysia (UKM)

Bangi, Selangor, Malaysia

anmar@ftsm.ukm.my

Masri Ayob

DMO/CAIT, Computer Science Department

Universiti Kebangsaan Malaysia (UKM)

Bangi, Selangor, Malaysia

masri@ftsm.ukm.my

Abstract— This work utilize Round Robin (RR) mechanism to

systematically explore neighbors of solution. RR is one of the

simplest scheduling algorithms, which assigns time slices to

each process in equal portions and in circular order handling

all processes without priority. In this work, we consider five

different neighborhood structures. RR gives each

neighborhood a certain number of iterations to explore some

neighbors of the current solution. Experimental results shows

that, our adaptive randomized descent algorithm using round

robin algorithm significantly produces good-quality solutions

(regarding Socha benchmark datasets), which outperformed

the ARDA approach and other single-based meta-heuristics.

Keywords-Course Timetabling Problems; Round Robin;

Neighborhood Selection.

I. INTRODUCTION

University course timetabling problems usually involves

assigning a set of students and a set of courses (events) to a

predetermined number of rooms and predetermined number

of timeslots [1]. The goal is to produce a feasible timetable

that satisfies all hard constraints under any circumstances

while satisfying as much as possible the soft constraints in

order to produce a good quality timetable. A good quality

timetable is indicated by the smaller number of soft

constraints violations (each violation of the soft constraints

will be penalized). A university course timetabling problem

has been classified as an NP-hard problem, which is

difficult to solve for optimality in a reasonable time [2].

Finding good quality solutions to these problems depends

on the technique itself and the structure employed during the

search [3].

Various approaches have been applied to solve

university course timetabling problems. Some of these

approaches are tabu search [4] simulated annealing [5],

randomized descent method [6], great deluge [7] and genetic

algorithms [8]. For further information on the previous

works, please refer to the following overview/ survey papers

[1] [3] [6].

Recently, a new approach on solving university exam

timetabling problems based on basic local search (late

acceptance strategy in hill climbing, LAHC) had been

proposed by Burke and Bykove [9]. The idea of LAHC is to

delay the comparison between the quality of the candidate

solution with a solution quality, which was “current” several

steps before the current solution. This is done by

maintaining a list of accepted solution’s quality (i.e. just

value of the penalty cost). The list is used as an acceptance

criterion. In our previous work, we applied LAHC over

university course timetabling problem [10] and extended the

work in [10] by introducing average late acceptance

randomized descent (ALARD) to solve university course

timetabling problem. However, both LAHC and ALARD

are descent heuristic, which may easily trap in local optima

(minima) [6], [9]. Hence, we extended the work in [10] by

proposing the adaptive randomized descent algorithm

ARDA [11], which overcomes the limitation of our previous

work [10]. However, since ARDA [11] employed composite

[12] neighborhood structure, it requires a proper mechanism

to manage the exploration of neighborhood structures.

Therefore, in this work, we utilize the round robin

algorithm as in [13] to systematically explore neighborhood

structures in order to enhance the performance of ARDA.

The aim of our work is to investigate the performance of

applying the adaptive average late acceptance randomized

descent algorithm using round robin algorithm for solving

university course timetabling problems. In order to evaluate

the effectiveness of the ARDA-RR, we made a comparison

between the performances of our ARDA-RR, RR previously

employed within dual sequence simulated annealing [13],

ARDA employing a composite neighborhood structures [11];

and other approaches applied over eleven standard

benchmark test datasets that were introduced by Socha et al.

[14].

II. PROBLEM DETAILS

In this work, the eleven standard benchmark test datasets

instances introduced by Socha et al. [14] are used, which

only tackle the students satisfaction for the university course

timetabling problem. The problem consists of:

• A set of Rooms R in which events can take place.

• A set of Events (courses) E to be scheduled in 45

timeslots (5 days of 9 hours each with one hour for

each timeslot).

• A set of Features F characterizing the rooms.

• A set of Students S who attend the events.

1201978-1-4244-8136-1/10/$26.00 c©2010 IEEE

These datasets are categorized into three groups, small (S1,

S2, S3, S4, S5), medium (M1, M2, M3, M4, M5) and large

(L); which their characteristics are described as in [14] (see

Table I). Table I also shows, the conflict density (CD) for

each dataset (representing the complexity), approximate

number of students enrolled in each event (Students/ Events),

and the approximate number of available rooms for each

event (Rooms/ Events) which are calculated as in [15].

These datasets were collected from various real-world

university course timetabling problems (see details

description in [14]). The problem consists of three hard

constraints (Hc1, Hc2 and Hc3) and three soft constraints

(Sc1, Sc2, and Sc3) as follows:

Hc1: No student can be assigned to more than one

event at the same time.

Hc2: The room capacity must be greater than or equal

to all the attending students and satisfy all the

features required by the event that assigned to it.

Hc3: No more than one event can take place in each

room at the same timeslot.

Sc1: A student should not have an event in the last time

slot of the day.

Sc2: A student should not have more than two events

consecutively.

Sc3: A student should not have a single event on a day.

To measure the quality of timetable, we use an

objective function as in [14], each violation of the soft or

hard constraints will be penalized ‘1’ for each student who is

involved in this situation [14].

The quality of the timetable is simply calculated as the

summation of the hard constraints violations hcv and the soft

constraints violations scv for each student (S equal the total

number of students), as shown in equation (1) by Rossi-

Doria et al. [16]:

(1)

In order to differentiate between feasible and infeasible

solutions, we multiply the violation of hard constraints by a

huge constant C, where C is a constant larger than the

possible number of soft constraint violations in order to

differentiate the infeasible solutions with feasible solution.

Apart from that, this will help in avoiding infeasible solution

as in [15].

III. ROUND ROBIN BASED ADAPTIVE AVERAGE LATE

ACCEPTANCE (ARDA-RR)

The round robin based adaptive average late acceptance

(ARDA-RR) is an improvement heuristic based on LAHC.

LAHC relies on a single parameter (the list length),

which allow some worsening moves [9], whilst, there is

other algorithms relies on a function such as great deluge

and simulated annealing [9] to allow some worsening

moves. The list in LAHC is used as the acceptance criterion

[9]. However, the limitation of LAHC is to decide a suitable

length of list L for each different benchmark problem [9].

Therefore, in our previous work, we proposed the

average late acceptance randomized descent (ALARD) [10],

which is based on LAHC idea, but uses an average value of

the values in the list of accepted solution as an acceptance

criterion [10].

However, both LAHC and ALARD are descent

heuristics (where they only accept solutions which are better

than or equal quality of the selected solution or average

quality, respectively). Thus, they may get trapped in local

optima. Therefore, we propose the adaptive average late

acceptance randomized descent [11]; ARDA (which

enhance the ALARD approach) to overcome the weakness

in LAHC and ALARD. As in ALARD, ARDA also use a

threshold value as an acceptance criterion. Apart from that,

ARDA can adaptively manage to escape from local optima

by adding the estimated value to the values in the list L

(which increases the threshold value). That is, we allow

some slightly worse solution to be accepted that might help

the algorithm to escape from local optima. However,

ARDA-RR need to manage the neighborhood exploration in

order to improve the quality of a solution.

==

+=

S

i

S

i

SscvCShcvSf1

*

1

** )(*)()(

TABLE I. ELEVEN DATASETS [14] [15]

Dataset #Students #Events #Rooms #Features Max

S/E

Max

E/S

Approx

F/R

%F

Usage CDStudents/

Events

Rooms/

Events

S1 80 100 5 5 20 20 3 0.7 10.96 4.98 0.82 S2 80 100 5 5 20 20 3 0.7 13.92 5.36 0.79

S3 80 100 5 5 20 20 3 0.7 9.71 4.65 1.00

S4 80 100 5 5 20 20 3 0.7 7.16 3.45 1.39 S5 80 100 5 5 20 20 3 0.7 15.10 5.99 1.17

M1 200 400 10 5 20 50 3 0.8 37.38 8.85 2.23

M2 200 400 10 5 20 50 3 0.8 37.66 8.84 1.91 M3 200 400 10 5 20 50 3 0.8 40.44 8.85 1.91

M4 200 400 10 5 20 50 3 0.8 37.50 8.81 1.88

M5 200 400 10 5 20 50 3 0.8 28.27 8.66 1.37 L 400 400 10 10 20 100 5 0.9 45.57 8.92 0.76

1202 2010 10th International Conference on Intelligent Systems Design and Applications

Therefore, in this work, we utilize the round robin

algorithm [13] [17] with adaptive average late acceptance in

randomized descent (ARDA-RR) to systematically manage

the exploration of neighborhood structures in order to

enhance the performance of ARDA. Fig.1 shows the pseudo

code for ARDA-RR for solving course timetabling

problems. In this work, we use round robin algorithm as in

[13] to select among five different neighborhood structures

(as in [10]). We manipulate three lists: one is used for storing

the accepted solutions quality (list L) for the purpose of

calculating the average value (acceptance criterion), the

second list is used to store the estimated values (list A); and

the third list is used to store each idle neighbourhoods with

the number of improvements obtained ( tabu list TL).

Figure 1. Pseudo code for ARDA-RR to solve course timetabling problems

Our approach starts with a given initial solution So, and

quality of So, as f(So). Let Sbest be the best obtained solution,

f(Sbest), be the quality of Sbest ; S* be the candidate solution,

f(S*), be the quality of S*; LLength be the length of the list L;

Avg be the average of the values in L; EV be the estimated

value stored in the list A; Idle be the ARDA-RR idle

Iterations.

ARDA-RR starts by initializing the required parameters

(see Step1, Fig.1) by assigning all elements in L equal to the

quality of the initial solution f(So) and set the average value,

Avg = f(So). In this work , we use a fix size of L (i.e. L size =

20).

In the improvement phase (Step2), we iteratively

improve the initial solution So until the stopping criteria is

met. At this phase, we generate some neighbor solutions (in

our case, randomly generate some “5” neighbors from a

selected neighborhood), and the best neighbor (S*) will be

selected to be the candidate solution (Step2.1). The

neighorhood is selected using round robin [17] selection

mechanism. The same neighborhood will be applied several

iterations in the improvement phase until the given quantum

end. In RR mechanism, each neighborhood will be given the

same quantum.

In our work, we consider five different neighborhood

structures as in [10]. Round Robin will give each

neighborhood a certain number of iterations to generate

neighbors of current solution until they are finished. In case

of improvement obtained during any of the quantum (three

iterations in our case), then we extend the quantum and

proceed with the same neighborhood. Otherwise, we

proceed to the next neighborhood in the sequence. If a

neighborhood obtained no solution improvement for a

number of successive quantum’s (e.g. 10 quantum in our

case), then the neighborhood structure will be removed from

the RR sequence and assigned to tabu list (TL). The tabu list

TL also store the frequency of improvement obtained by the

neighbourhood structure. When all neighborhood structures

are move into TL , then the new sequence for RR will be

rearrange based on the frequency improvement obtained by

neighborhoods structure (starting with the highest

frequency first).

At each iteration if the candidate solution S*better than

the best solution Sbest, we will accept the solution S*(in

Step2.2, i.e. So = S*), update the frequency of EV in the list

A, update f(Sbest) = f(S*) and sort the element in descending

order based on their frequency.

Or if f(S*) is just better than Avg, we will only accept f(S*)

and replace the element value that was the longest time in L

with the quality of the new accepted solution. We then

recalculate Avg.

Otherwise, the S* will be rejected. We then increase the

idle iteration by one, updating So= Sbest and proceed with the

next iteration.

In Step2.3, we set the estimated value, EV equals to the

first value in A (i.e. the one with the highest marked of

repetition), when number of idle iterations equals to the

Step 1: Initialization Phase

Determine initial candidate solution So and f(So)

Set Idle = 0; Sbest=So; f(Sbest)= f(So);

Set L Values = f(So); Avg=average of L values;

Step 2: Improvement (Iterative) Phas

while termination condition is not satisfied :

Step 2.1: Selecting candidate solution S*

Select the neighborhood structure, Ni using round

robin approach, or continue apply the same Ni

until the quantum end.

Generate some neighbors of So in Ni and select

the best neighbor, S*.

Step 2.2: Accepting Solution

if f(S*) < f(Sbest)

Idle =0; EV= f(So) - f(S*);

Sbest:=S*; f(Sbest)=f(So);

So=S*;

Update the frequency of EV in the A (or add EV

in A if it is not exist).

Sort the EV elements in A in descending order

based on their frequency.

elseif f(S*) < Avg // good solution

Idle =0; So= S*;

Replace the element value that was the longest

time in L with f(S*);

Recalculate Avg;

else // bad solution

So= Sbest; Idle = Idle + 1;

end if

Step 2.3: Idle Iteration

if Idle >= Maximum number of Idle iterations

EV = A [0]; // get the first element in A

Add all values in L with EV;

Recalculate Avg;

Rotate left all elements in A;

end if

end while

Step 3: Termination phase

Return the best found solution Sbest

2010 10th International Conference on Intelligent Systems Design and Applications 1203

maximum number of idle. Then, we update all values in L

(by adding EV to the values), update all values in A (by

rotate left all elements in A) and recalculate Avg.

In the termination phase Step3, we return the best

solution found Sbest , after we iteratively improve the initial

solution So until the stopping condition is met (Step2).

ARDA-RR can adaptively manage to escape from local

optima by the corporation between the systematic selection

of neighborhoods using round robin algorithm [17]; and

intelligently increase the threshold value when the search

trap in local optima (i.e. when the generated solutions quality

from the neighborhoods are worse ‘idle’, or the threshold

reach plateau). This is done by estimating an appropriate

threshold value based on history of search. In other words,

ARDA-RR intelligently helps to prevent from getting

trapped in local optima by the corporation between adaptive

acceptance criterion and manipulates a round robin

algorithm, which is used to control the selection of

neighborhood structures while maintaining variations of

neighborhood sequence.

Our RR configuration concerns of providing an

interaction between controlling neighborhood selection and

sequence the acceptance criterion (average). In this sense,

this configuration is quite similar to the work presented in

[13] basic idea. In the other hand, our RR differs from the

respected work in [13] in term of using a number of

iterations (quantum) rather than using time slice. Our work

also differs from [13], where we switch between

neighborhoods in both cases of improvement and non

improvement after a predefined number of iterations. Whilst

in [13], they switch between neighborhoods only in case of

non improvement, where, a neighborhood is employed until

it obtains no improvement of a solution. Our work also

differs from [13], that the mechanism of switching between

neighborhoods sequence is ordered, in our work once no

solution improvement obtained for a number of successive

iterations for all neighborhoods, then the sequence is

ordered based on the highest improvement neighborhood

(descending order) in term of solution quality; whilst in

[13], they use a fixed order sequence. The illustrated

behavior above has been determined by our preliminary

experiment.

IV. EXPERIMENTAL RESULTS

In this work, we generate initial solution by using

constructive heuristic that was introduced by Landa-Silva

and Obit [18]. We have run our algorithm 11 times across

11 instances.

Our algorithm stops after 200,000 iterations (based on

literature and our preliminary experiments). Our algorithm’s

parameters setting were determined experimentally as

shown in Table II. The algorithm was tested on a PC with

an Intel dual core 1800 MHz, 1GB RAM. Table III shows

experimental results for the comparison between ARDA-RR

using our proposed round robin algorithm, ARDA

employing a composite neighborhood structure [11];

ARDA-RR using round robin algorithm proposed in [13],

LAHC, ALARD, and other meta-heuristic searches that

were tested on Socha benchmark datasets. The best results

are presented in bold.

TABLE II. PARAMETERS SETTINGS USED IN OUR ALGORITHM

Parameter Value

MI Termination condition (Number of Iterations) = 200000

Llength The length of List L “store the accepted solution quality”

= 20

Idle Maximum iterations number of ARDA-RR to consider idle = 10

RDI Randomized descent improvement iterations = 5

RRq Round Robin Iterations (quantum) = 3

Idle RRq Idle of Round Robin neighbourhoods structure = 10 quantum

TABLE III. COMPARISON BETWEEN OUR METHODOLOGY AND OTHER LOCAL HYBRID META-HEURISTIC SEARCH

RESULTS ON COURSE TIMETABLING PROBLEM

Note: L1. TB-MA[19] L2. MHSA[20] L3. HPCA[21] L4. NLGDHH-SD[22] L5. HEA[23] L6. EGD[24] L7. DSSA[13] L8. ALARD[10] L9. LAHC[10] L10. ARDA [11] L11. ARDA using RR proposed in [13]

Dataset Min Avg Std.Dev L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

S1 0 0.64 1.03 0 0 0 0 0 0 0 0 0 0 0

S2 0 0.73 1.01 0 0 0 0 0 0 0 0 0 0 0

S3 0 1.27 1.01 0 0 0 0 0 0 0 0 0 0 0

S4 0 1.27 0.90 0 0 0 0 0 0 0 0 0 0 0

S5 0 0.64 0.81 0 0 0 0 0 0 0 0 0 0 0

M1 51 63.91 9.25 55 168 77 71 221 80 93 143 153 82 64

M2 54 64.45 7.63 70 160 73 82 147 105 98 130 137 78 58

M3 95 108.45 9.29 102 176 133 137 246 139 149 183 207 136 107

M4 48 64.18 10.66 32 144 69 55 165 88 103 133 142 73 51

M5 75 90.18 9.77 61 71 101 106 130 88 98 169 186 103 77

L 609 627.73 12.35 653 417 627 777 529 730 680 825 863 680 623

1204 2010 10th International Conference on Intelligent Systems Design and Applications

Results in Table III shows, that our proposed round

robin outperformed the ARDA in [11] (L10) and round

robin proposed by [13] (L11) in all datasets. Results in

Table III also shows, that ARDA-RR outperformed LAHC

(L9), ALARD (L8) and HPCA (L3) (which is the best

single based search known result until today) in all datasets.

Meanwhile, ARDA-RR results of all small datasets scored

zero same as the best results obtained by other approaches

in the literature. ARDA-RR results for medium1, medium2

and medium3 datasets have outperformed the best known

results obtained by other approaches in the literature.

Whereas, Turabieh and Abdullah [19] (L1) in 2009

outperformed many other approaches in the literature (with

regards to Socha benchmark datasets) for medium4 and

medium5 datasets. Whereas Al-Betar et al. [20] (L2) in

2010 scored the best result until today for the large dataset. .

Fig.2 shows the box and whisker plot details of the ARDA-

RR results for 11 runs.

Figure 2. Box and whisker plot of ARDA-RR for all datasets

Fig.2 shows, the box and whisker plot that summarize

the results of 11 runs for each dataset by ARDA-RR

algorithm on Socha benchmark datasets. For all small,

medium3, medium 4, and large datasets, we can see that the

median is more close to the best than worst of these runs.

Meanwhile, the rest of the medium datasets considered as

normal. This indicates that the algorithm is stable and

consistent and most of the time can produce very good

quality solution.

V. CONCLUSIONS AND DISCUSSION

The overall goal of this work is to investigate the

effectiveness of the new strategy proposed to intelligently

control neighborhoods selection, namely as round robin

algorithm incorporated into an adaptive average late

acceptance randomized descent algorithm (ARDA-RR) to

solve course timetabling problems. The RR implementation

enables our approach (ARDA) in [11] to explore the

neighbors of a solution more effectively.

In this work, we employed five different neighborhood

structures sequentially using RR that gives each

neighborhood a certain number of improvement iterations to

generate neighbors of current solution until they are all

explored. This gives each neighbor the chance to be

explored to speed up the search. In case of improvement

obtained during any of the quantum (three iterations), then

we extend the quantum and proceed with the same

neighborhood. If no solution improvement encountered in a

predetermined number of successive non improvement

iterations, then this neighborhood will be removed from the

sequence and located in a list (short term memory). The

sequence form could be changed after no solution

improvement obtained for a number of successive iterations

for all neighborhoods. The new sequence will order all the

neighborhoods from TL (short term memory) starting from

the highest improvement neighborhood.

In order to evaluate the effectiveness of ARDA-RR for

solving the university course timetabling problem, we test

ARDA-RR on Socha benchmark dataset [14]. Results

indicates that ARDA-RR approach is capable of producing

good-quality solutions with outperformed other single based

approaches in all datasets results and outperformed

population based approaches in some datasets results in the

literature; which make it particularly suitable for solving

course timetabling problems. Our future work is to improve

ARDA-RR by hybridize it with other approach.

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