Einführung

1. Einführung University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Round Robin Tournaments and Three Index Assignment Dirk Briskorn briskorn@bwl.uni-kiel.de Joint Work with Andreas Drexl, Frits C.R. Spieksma 1

2. Einführung Outline University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • Round Robin Tournaments • Min Cost Single RRT Problem • Min Cost Double RRT Problem • Planar Three Index Assignment Problem • Complexity • Min Cost DRRT Problem as TIAP • Conclusion and Outlook 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

3. Einführung Single Round Robin Tournaments University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • league of n teams (|T| = n even) • each pair of teams meets exactly once • each team plays exactly once per matchday • |P| = n-1 matchdays • n(n-1)2 possible matches • n(n-1)/2 arranged matches 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

4. Einführung Single Round Robin Tournaments University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • (1): minimization of cost • (2): each pair of teams meets exactly once • (3): matches form matchdays 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

5. Einführung Double Round Robin Tournaments University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • league of n teams (|T| = n even) • each pair of teams meets exactly twice (once at each venue) • each team plays exactly once per matchday • |P| = 2n-2 matchdays • 2n(n-1)2 possible matches • n(n-1) arranged matches 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

6. Einführung Double Round Robin Tournaments University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • (2): each pair of teams meets exactly once at both venues • (3): matches form matchdays 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

7. Einführung Three Index Assignment Problem University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • 3 disjunctive sets TH, TA, P of cardinality n • cost for each triple in (TH X TA X P) • find minimum costtriples such that each tuple in (THX TA), (THX P), and (TAX P) is contained exactly once • n3 possible triples • n2 chosen triples 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

8. Einführung Three Index Assignment Problem University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • (1): minimization of cost • (2) – (4): each tuple is part of a chosen triple 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

9. Einführung Reducing TIAP to SRRT University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Solution to TIAP Solution to SRRT 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

10. Einführung Reducing TIAP to SRRT University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Solution to TIAP of size n as Part of a Solution to SRRT problem of size 2n 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

11. Einführung Reducing TIAP to SRRT University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Complement of solution to SRRT problem of size 2n 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

12. Einführung Reducing TIAP to SRRT University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • instantiation of min cost SRRT problem • n‘ = 2n • c‘: 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

13. Einführung Reducing TIAP to SRRT University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Reduction Scheme from TIAP to min cost SRRT problem 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

14. Einführung Reducing DRRT to TIAP University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Home-Away-Pattern Set Solution to TIAP 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

15. Einführung Reducing DRRT to TIAP University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • instantiation of TIAP • n‘ = n • c‘: 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

16. Einführung Conclusion and Outlook University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics • min cost SRRT problem is NP-hard • special case of min cost DRRT problem is reducable to TIAP • close relation of problems‘ structures • mechanism to translate RRTs into TIAPs • modification of assignment approaches into RRT scheduling techniques 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn

17. Einführung Conclusion and Outlook University of Kiel Wirtschafts- und Sozialwissenschaftliche Fakultät Institute for Production and Logistics Thanks for your Interest! Any Questions? 1 Scheduling Algorithms for New Emerging Applications – May/June 2006 - Dirk Briskorn