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1 Fuzzy Multiple Heuristic Ordering for Course Timetabling Hishammuddin Asmuni Edmund K. Burke Jonathan M. Garibaldi UKCI2005 – London, UK Automated Scheduling, optimisAtion and Planning (ASAP) Group, School of Computer Science and IT University of Nottingham Jubilee Campus, Wollaton Road Nottingham, NG8 1BB, UK

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

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3 Course Timetabling Problem MondayTuesdayWednesdayThursdayFriday S1C1 S2C2 S3C3 S4C4 S5C5....... CN STST Time slot 1 Time slot 2 C1C2C3C4 C1 S1S1S3S3 C2 S1S1S4S4 C3 S1S3S1S4S3S4 C4 S3S4S3S4 Assign the set of events to time slots, subjects to specified constraints Which event should be scheduled first ? randomly based on how difficult to schedule the event

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4 Graph Based Heuristic Ordering C1C2C3C4 C1 95 C2 7 C3 93 C4 573 C1 C3 C2 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 Heuristic use to measure event difficulties to be scheduled: C4 C1 C3 C2

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5 Graph Based Heuristic Ordering C1C2C3C4 C1 95 C2 7 C3 93 C4 573 C1 C3 C2 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. C4 C1 C3 C2

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6 Graph Based Heuristic Ordering C1C2C3C4 C1 95 C2 7 C3 93 C4 573 C1 C3 C2 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. C4 C1 C3 C2

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7 Graph Based Heuristic Ordering C1C2C3C4 C1 95 C2 7 C3 93 C4 573 C1 C3 C2 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). C4 C1 C3 C2

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8 Graph Based Heuristic Ordering C1C2C3C4 C1 95 C2 7 C3 93 C4 573 C1 C3 C2 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. C4 C1 C3 C2

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9 Fuzzy Modeling Choose heuristics combination from heuristic list – SD, LD, LE, wLD and LCD Generate fuzzy rules that related to heuristics chosen. Define fuzzy membership functions for each heuristic Sequential Constructive Algorithm Calculate events difficulty to be scheduled Problem Definitions Constructive Initial Solution Iterative improvement Optimal Solution Assign event to timeslot Anymore events? Reorder events Ordered events with decreasingly difficulty Yes No General Framework Non-fuzzy single heuristic ordering

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10 Rescheduling scheduled events 1 2 3 4... P Unscheduled events Insert event e x List of events that conflict with e x Bump back event move event to other timeslot

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11 Sequential Constructive Algorithm Fuzzy Modeling Choose heuristics combination from heuristic list – SD, LD, LE, wLD and LCD Generate fuzzy rules that related to heuristics chosen. Define fuzzy membership functions for each heuristic Calculate events difficulty to be scheduled Problem Definitions Constructive Initial Solution Iterative improvement Optimal Solution Assign event to timeslot Anymore events? Reorder events Ordered events with decreasingly difficulty Yes No Fuzzy Multiple Heuristic Ordering Non-fuzzy Single Heuristic Ordering

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12 Fuzzy Model – Fixed Fuzzy Rules LELD SMH SD SMHSMHSMH SSVS SS MSS MSS HMMHMM HHSSHMMVHHM S - Small M - Medium H – High VS – Very Small VH – Very High If LD is High and SD is Small and LE is High then examweight is Very High

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13 Fuzzy Model - Membership Function TUNING !

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14 Problem Definition (1) : Data set Dataset No of courses No of Students No of rooms Small1 100805 Small2 100805 Small3 100805 Small4 100805 Small5 100805 Medium1 40020010 Medium2 40020010 Medium3 40020010 Medium4 40020010 Medium5 40020010 Large 400 10

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15 Problem Definition (2) 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 Task - Assign courses to time slots that must satisfy the following hard constraints :

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16 Problem Definition (3) 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 Objective - Minimise violation of the following soft constraints :

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17 Experimental Results : Comparison of solution quality Single Heuristic Ordering DatasetBest FuzzyLDSDLCDLEWLD Small1 107831487980 Small2 94544553452 Small3 72830424127 Small4 174250485148 Small5 74129744347 Medium1 243423345433465445 Medium2 325-398--- Medium3 249-298--- Medium4 285-403--- Medium5 132296252307399445 Large 1138----- - : infeasible solution

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18 Experimental Results : Comparison of number of rescheduling procedures Single Heuristic Ordering DatasetBest FuzzyLDSDLCDLEWLD Small1 000000 Small2 000000 Small3 000000 Small4 000000 Small5 000000 Medium1 04001226059 Medium2 0200*0 Medium3 0200*0 Medium4 1200*0 Medium5 020514140 Large 307500* * : maximum number of rescheduling procedures allowed

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19 Experimental Results : Comparison with other methods Dataset Our Best Results Graph Based Hyper Heuristic VNS with Randomized Improvement Tabu Search Hyper Heuristic Local Search Ant Algorithm Small1 10 60181 Small2 9 702113 Small3 7 30081 Small4 17 30171 Small5 7 40050 Medium1 243 372242146199195 Medium2 325 419161173202.5184 Medium3 249 359265267-248 Medium4 285 348181169177.5164.5 Medium5 132 171151303-219.5 Large 1138 1068-1166-851.5

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20 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

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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

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22 Than k You UKCI2005 – London, UK Fuzzy Multiple Heuristic Ordering for Course Timetabling

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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; and wd, we, ws are weighting factors for LD, LE and SD respectively.

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24 Heuristic Ordering : single vs multiple LDLE e13040 e21030 e35020 e42035 e53910 e61043 e71020 e81925 e92715 e104530 LDLE e35020 e104530 e53910 e13040 e92715 e42035 e81925 e21030 e61043 e71020 LDLE e61043 e13040 e42035 e21030 e104530 e81925 e71020 e35020 e92715 e53910 LDLEweight e1045300.7 e350200.6 e130400.55 e539100.52 e420350.5 e610430.5 e927150.47 e819250.47 e210300.4 e710200.35 unordered ordered by LD ordered by LE ordered by weight 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

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25 Fuzzy Inference Graphics representation: w2 T-norm w1 C weight z is z COA LDLELD is 7 LE is 35 Crisp value high weight mediumhigh LDLE Rule 1If LD is medium AND LE is high then weight is high fast weight smallmedium LDLE Rule 2If LD is small AND LE is medium then weight is medium

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26 Best Membership Functions CAR-F-92 CAR-S-91 HEC-S-92

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27 Best Membership Functions (continue) STA-F-83 UTA-S-92 YOR-F-83

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