# Tuning and Optimization of A Fuzzy Model Using A

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UKCI2005 – London, UK

Fuzzy Multiple Heuristic Ordering
for Course Timetabling

Hishammuddin Asmuni
Edmund K. Burke
Jonathan M. Garibaldi

Automated Scheduling, optimisAtion and Planning (ASAP) Group,
School of Computer Science and IT
University of Nottingham
Nottingham, NG8 1BB, UK
2

Outline

• Introduction to Timetabling Problem
• Graph Based Heuristic Ordering
• Sequential Construction Algorithm
• Fuzzy Modeling
• Experimental Results
• Problem Definitions
• Results
• Conclusions and Future Work
3

Course Timetabling Problem

Monday        Tuesday    Wednesday   Thursday    Friday
Time slot 1
Time slot 2

S1      C1
S2      C2         Assign the set
of events to
S3      C3         time slots,
S4      C4         subjects to
specified
S5      C5         constraints
C1     C2         C3    C4
.       .      Which event                  C1            S1    S1S3      S3
should be
.       .                                         S1            S1S4      S4
scheduled first ?            C2
.       .          • randomly
C3   S1S3    S1S4            S3S4
.       CN         • based on how
difficult to schedule
C4    S3      S4    S3S4
ST                 the event
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Graph Based Heuristic Ordering
C1    C2   C3    C4       C1           C2
C1               9     5

C2                     7

C3    9                3

C4    5     7    3             C3           C4

Heuristic use to measure event difficulties to be scheduled:
Largest Degree (LD) First

The degree of an event is simply a count of the number of
other events which conflict in the sense that students are
enrolled in both events. This heuristic orders events in
terms of those with the highest degree first
5

Graph Based Heuristic Ordering
C1    C2    C3   C4       C1            C2
C1                9    5

C2                     7

C3    9                3

C4    5     7     3            C3            C4

Heuristic use to measure event difficulties to be scheduled:

Weighted Largest Degree (WLD) First

This heuristic also based on LD. Beside the number of
events in conflict, the total number of students involved in
the conflict are taken into account as well.
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Graph Based Heuristic Ordering
C1    C2    C3    C4      C1            C2
C1                9    5

C2                     7

C3    9                3

C4    5     7     3             C3           C4

Heuristic use to measure event difficulties to be scheduled:
Largest Coloured Degree (LCD) First

This heuristic is based on LD. For this heuristic, only
events which already assigned to the schedule are
considered as the events which will cause conflict.
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Graph Based Heuristic Ordering
C1    C2   C3    C4       C1           C2
C1               9     5

C2                     7

C3    9                3

C4    5     7    3             C3           C4

Heuristic use to measure event difficulties to be scheduled:
Largest Enrollment (LE) First

The number of students enrolled for each event is used to
order the events (the highest number of student first).
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Graph Based Heuristic Ordering
C1    C2    C3   C4       C1           C2
C1               9     5

C2                     7

C3    9                3

C4    5     7    3             C3           C4

Heuristic use to measure event difficulties to be scheduled:
Least Saturation Degree (SD) First

The number of time slots available is used to order the
events. The basic motivation is that events with less time
slots available are more likely to be difficult to be
scheduled. The fewer time slots that are available, the
higher up the ordering is the event.
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General Framework
Sequential Constructive AlgorithmModeling
Fuzzy
Choose heuristics combination from
heuristic list – SD, LD, LE, wLD and LCD
Non-fuzzy single
heuristic ordering
Generate fuzzy rules that related to
heuristics chosen.

Define fuzzy membership functions for
each heuristic                  Problem
Calculate events difficulty to                                  Definitions
be scheduled

Constructive
Initial Solution
Ordered events with
decreasingly difficulty
Iterative improvement
Assign event to timeslot

Reorder                                              No                          „Optimal‟ Solution
Yes             Anymore
events
events?
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Rescheduling scheduled events

Insert event ex
1      2   3       4      . . .   P

move event
Unscheduled                 to other
events                      timeslot

List of
events
that
conflict
Bump back event                    with ex
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Fuzzy Multiple Heuristic Ordering
Sequential Constructive Algorithm               Fuzzy Modeling
Choose heuristics combination from
Non-fuzzy Single                        heuristic list – SD, LD, LE, wLD and LCD
Heuristic Ordering
Generate fuzzy rules that related to
heuristics chosen.

Define fuzzy membership functions for
each heuristic

Problem
Calculate events                                                      Definitions
difficulty to be scheduled

Constructive Initial
Solution
Ordered events with
decreasingly difficulty
Iterative improvement
Assign event to timeslot

Yes                                         No
Reorder                                                                                  „Optimal‟ Solution
Anymore
events
events?
12

Fuzzy Model – Fixed Fuzzy Rules

LE                    LD
S            M            H
S - Small
SD           SD           SD       M - Medium
S   M    H   S   M    H   S   M    H   H – High
S    S VS VS S        S VS M       S    S   VS – Very Small
VH – Very High
M    S   S    VS H    M    M   H   M    M
H    H   S    S   H   M    M VH H       M

If LD is High and SD is Small and LE is High
then examweight is Very High
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Fuzzy Model - Membership
Function
small             medium                       high
1

medium                           high
1
0.5

0.5
0
0                0.2     0.4        0.6           0.8            1
cp                 small                   medium                0 high
1                                                       0             0.2    0.4    0.6    0.8          1

0.5
TUNING !                           cp

0
0            0.2           0.4        0.6    0.8              1
cp
small          medium                             high                                      small                            medium
1                                                                                          1

0.5                                                                                            0.5

0
0
0                                                                1                          0           0.2       0.4    0.6    0.8       1

cp
cp
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Problem Definition (1) : Data set
No of      No of     No of
Dataset
courses   Students   rooms
Small1       100       80         5
Small2       100       80         5
Small3       100       80         5
Small4       100       80         5
Small5       100       80         5
Medium1      400       200       10
Medium2      400       200       10
Medium3      400       200       10
Medium4      400       200       10
Medium5      400       200       10
Large        400       400       10
15

Problem Definition (2)
Assign courses to time slots that must satisfy the
following hard constraints :
1. No student is required to attend more than one
course at the same time
2. A course can only be scheduled to a room which
satisfies the features required by the course
3. A course can only be scheduled to a room which
has enough room to accommodate all students
registered for it
4. Only one course can be scheduled in one room at
any time slot
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Problem Definition (3)

Objective -

Minimise violation of the following soft constraints :
1. No student should be scheduled to attend only
one course on a day
2. No course should be scheduled at the last time
slot of the day for any student
3. No student should be scheduled to attend more
than two courses consecutively in any one day
Experimental Results :
17

Comparison of solution quality
Single Heuristic Ordering

Dataset    Best Fuzzy   LD      SD     LCD          LE      WLD

Small1        10        78       31      48          79       80
Small2         9        45       44      55          34       52
Small3         7        28       30      42          41       27
Small4        17        42       50      48          51       48
Small5         7        41       29      74          43       47
Medium1       243       423     345     433        465       445
Medium2       325         -     398       -           -         -
Medium3       249         -     298       -           -         -
Medium4       285         -     403       -           -         -
Medium5       132       296     252     307        399       445
Large        1138         -       -       -           -         -
-   : infeasible solution
Experimental Results : Comparison of
18

number of “rescheduling procedures”
Single Heuristic Ordering

Dataset   Best Fuzzy      LD      SD      LCD       LE      WLD

Small1        0             0       0        0       0        0
Small2        0             0       0        0       0        0
Small3        0             0       0        0       0        0
Small4        0             0       0        0       0        0
Small5        0             0       0        0       0        0
Medium1       0           40        0     122       60       59
Medium2       0          200*       0     200*    200*      200*
Medium3       0          200*       0     200*    200*      200*
Medium4       1          200*       0     200*    200*      200*
Medium5       0             2       0       51      41       40
Large        307         500*    500*     500*    500*      500*
* : maximum number of “rescheduling procedures” allowed
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Experimental Results :
Comparison with other methods
Graph Based     VNS with    Tabu Search
Our Best                                              Local      Ant
Dataset                 Hyper       Randomized      Hyper
Results                                              Search   Algorithm
Heuristic    Improvement    Heuristic

Small1         10         6             0             1           8         1

Small2          9         7             0             2          11         3

Small3          7         3             0             0           8         1

Small4         17         3             0             1           7         1

Small5          7         4             0             0           5         0

Medium1        243       372           242           146         199       195

Medium2        325       419           161           173        202.5      184

Medium3        249       359           265           267          -        248

Medium4        285       348           181           169        177.5     164.5

Medium5        132       171           151           303          -       219.5

Large         1138       1068            -           1166         -       851.5
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Conclusions

•Better solutions can be produced when events are
ordered by several heuristic ordering simultaneously

•Tuning the fuzzy model can improve the performance

•No generic fuzzy model that suits all the datasets
21

Future Work

• investigating other combinations of heuristic ordering
• investigating different sets of fuzzy rules and fuzzy
membership functions
• exploring the use of more sophisticated optimization
algorithms to search for optimal fuzzy model
22

UKCI2005 – London, UK

Fuzzy Multiple Heuristic Ordering
for Course Timetabling

Than
k
You
23

Linear Weighting

W(ej) = wdLDj + weLEj + wsSDj

where j = 1,2, . . . N; wd= we = ws= {0.0, 0.1, …, 1.0} if
N <= 400; or wd= we= ws= {0.0, 0.25, 0.5, 0.75, 1.0} if
N > 400; and wd, we, ws are weighting factors for LD,
LE and SD respectively.
24

Heuristic Ordering : single vs multiple
Heuristic ordering use to measure the difficulty to schedule an event:
• largest degree (LD) - number of other events in conflict
• Largest enrolment (LE) - number of students enrolled
• Saturation degree (SD) - number of clash free timeslots available

LD    LE         LD   LE           LD   LE                  LD   LE   weight

e1     30   40   e3    50   20     e6    10   43            e10   45   30      0.7

e2     10   30   e10   45   30     e1    30   40            e3    50   20      0.6

e3     50   20   e5    39   10     e4    20   35            e1    30   40     0.55

e4     20   35   e1    30   40     e2    10   30            e5    39   10     0.52

e5     39   10   e9    27   15     e10   45   30            e4    20   35      0.5

e6     10   43   e4    20   35     e8    19   25            e6    10   43      0.5

e7     10   20   e8    19   25     e7    10   20            e9    27   15     0.47

e8     19   25   e2    10   30     e3    50   20            e8    19   25     0.47

e9     27   15   e6    10   43     e9    27   15            e2    10   30      0.4

e10    45   30   e7    10   20     e5    39   10            e7    10   20     0.35

unordered         ordered           ordered                        ordered
by LD             by LE                           by
weight
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Fuzzy Inference
Graphics representation:

Rule 1    If LD is medium AND LE is high then weight is high
medium        high                 high
w1
weight
LD                 LE
Rule 2 If LD is small AND LE is medium then weight is medium
small              medium                   fast
w2
weight
LD                 LE
T-norm

Crisp                                              C’
value LD is 7     LD     LE is 35    LE                   weight
z is zCOA
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Best Membership Functions
CAR-F-92
medium                                                         medium                                                 high                         small                   medium                                    high
small                                                                                          1
small                                                                                          1
1

0.5                                                                                                      0.5                                                                                                        0.5

0                                                                                                      0                                                                                                      0
0            0.2             0.4          0.6                0.8                   1                    0            0.2                  0.4                    0.6              0.8            1              0               0.2                    0.4      0.6             0.8                1
saturation degree                                                                                                    largest enrollment                                                                                examw eight

CAR-S-91
medium                                        high                                                                                  high                                        medium                               high                      small                  medium                high
small                            medium                                       1
small                                                               1
1                                                                          1

0.5                                                                        0.5                                                                                   0.5                                                                           0.5

0                                                                         0                                                                                      0                                                                               0
0          0.2         0.4            0.6   0.8            1               0           0.2            0.4         0.6           0.8             1                0            0.2           0.4       0.6          0.8          1             0             0.2   0.4         0.6         0.8              1
largest degree                                                                 saturation degree                                                                  largest enrollment                                                           examw eight

HEC-S-92
small                        medium                 high                       small         medium                                    high                           small                         medium               high                      small                     medium                high
1                                                                             1                                                                                  1                                                                               1

0.5                                                                         0.5                                                                                  0.5                                                                           0.5

0                                                                              0                                                                                 0                                                                                0
0             0.2         0.4            0.6   0.8            1               0           0.2            0.4          0.6           0.8                1            0            0.2           0.4      0.6       0.8              1             0             0.2   0.4         0.6         0.8              1
largest degree                                                                     saturation degree                                                             largest enrollment                                                            examw eight
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Best Membership Functions (continue)
STA-F-83
small                          medium            high                         small                            medium          high                        small                              medium          high                     medium                               high
1                                                                         1                                                                            1                                                                           1

0.5                                                                          0.5                                                                          0.5                                                                         0.5

0                                                                        0                                                                            0                                                                           0
0           0.2            0.4        0.6    0.8           1             0               0.2      0.4    0.6            0.8             1             0               0.2       0.4        0.6            0.8           1         0                0.2    0.4    0.6    0.8             1
largest degree                                                        saturation degree                                                                 largest enrollment                                                  examw eight

UTA-S-92
small         medium                            high                         small                         medium            high                        small          medium                              high                      medium                                high
1                                                                            1                                                                           1                                                                            1

0.5                                                                      0.5                                                                          0.5                                                                          0.5

0                                                                         0                                                                            0                                                                              0
0           0.2            0.4        0.6    0.8           1                 0           0.2       0.4    0.6            0.8                1            0            0.2       0.4            0.6        0.8           1             0             0.2    0.4    0.6    0.8                1
largest degree                                                         saturation degree                                                                 largest enrollment                                                      examw eight

YOR-F-83
small   medium                                  high                         small         medium                          high                           small                   medium                    high                       small                        medium         high
1                                                                        1                                                                         1                                                                               1

0.5                                                                      0.5                                                                          0.5                                                                          0.5

0                                                                        0                                                                            0                                                                           0
0         0.2            0.4        0.6    0.8              1             0           0.2       0.4    0.6            0.8                1            0            0.2       0.4            0.6        0.8           1             0             0.2    0.4    0.6    0.8                1
largest degree                                                        saturation degree                                                                 largest enrollment                                                      examw eight

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