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Optimization Problems in Round Robin Tournaments and Assignment Challenges

This document presents an in-depth exploration of various scheduling and optimization problems associated with Round Robin Tournaments (RRT) and the Three-Index Assignment Problem (TIAP). It outlines the minimal cost single and double RRT problems, detailing their complexities and the methodologies to solve them. Additionally, it discusses the intricacies of minimizing costs in tournament scheduling and presents reduction strategies from TIAP to simpler problems. The insights aim to enhance understanding of these algorithmic challenges within production and logistics frameworks.

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Optimization Problems in Round Robin Tournaments and Assignment Challenges

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  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

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