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  1. Sports Scheduling An Assessment of Various Approaches to Solving the n-Round Robin Tournament Noren De La Rosa Mallory Ratajewski

  2. Introduction • Scheduling tournaments in a sports league • Focusing on Round Robin Tournaments • Single Round Robin Tournament (SRRT): each team meets every other team once • Double Round Robin Tournament (DRRT): each team meets every other team twice • Economic impact: quality of schedule affects the revenue of the sports team

  3. What is the best schedule? • Home-away pattern • Balance between number of home and away games • Prefer alternating home away pattern • Any deviation is considered a break • Minimizing Distance Travelled • Other factors: • Availability of stadium • Preferences to increase revenues • Top team and bottom team constraints • Geographical constraints

  4. Main Considerations • Minimizing number of breaks • Used when teams return home after each away game instead of travelling to another away game • Alternating patterns usually preferred • Considers the fans • Ensures regular earnings from home games • Minimizing distance travelled • Used when teams travel to multiple away games without returning home • Huge savings can be obtained

  5. Minimizing Number of Breaks • Graph theoretical approaches • 1-factorization: partitioning the games into n-1 slots, each node will be incident to exactly one edge in each 1-factor • Practical applications: constrained minimum break problem • Decomposition approach • Combinatorial design, IP, enumeration techniques • See Nemhauser and Trick • Constraint programming approaches; see Henz

  6. Minimizing Distance Travelled • Similar to a travelling salesman problem • Problem is too large to solve using IP in a reasonable amount of time • Various heuristics used instead • The travelling tournament problem proposed by Easton, Nemhauser, and Trick

  7. The Travelling Tournament Problem • Double round robin tournament to be played by n teams over (2n-2) periods or weeks, where each team plays every period • Objective: minimize distance travelled by each team • Additional Constraints: • Maximum “road trip” of three games • Maximum “home stand” of three games • Repeater rule

  8. TTP Solution Approaches • What Methods Can We Use to Solve the TTP? • Integer programming • Constraint programming • Hybrid approaches involving heuristics

  9. A Tiling Approach for Fast Implementation of the TTP • Model the road trips as “tiles” • Each tile will contain “blocks”, which represent individual games • (i.e. – a road trip with 3 opponents is considered as one tile, with 3 blocks) • Three phase approach: • Phase I – Tile Creation • Phase II – Tile Placement • Phase III – Block Placement

  10. TTP Tiling Algorithm • Create a set of tiles for each team. These tiles are placed in a grid of n rows representing teams and (2n-2) columns representing weeks • As tiles are placed, other cells of the grid are filled in to keep the schedule consistent • When there are no tiles remaining, they are broken into their component blocks • If not all the blocks can be placed  block placement is backtracked to find additional solutions • If all blocks can be placed  a solution is generated

  11. TTP Tiling Algorithm Scheduling grid and tiles for Team 1 & Team 2

  12. A Demonstrative Example

  13. Create Tiles Find the MST from Prim’s algorithm C P B Y H D Y D H P B Phase I: Tile Creation

  14. Phase II: Tile Placement

  15. Phase III: Block Placement • All remaining unplaced tiles are broken into individual blocks • These blocks are placed into the scheduling grid • Backtracking is used when blocks do not lead to a solution

  16. Conclusions • Sports scheduling has huge economic implications for the sports industry • Optimal solutions that consider the many constraints are time consuming • Hybrid solutions involving heuristics are close to optimality and require less time • Many opportunities for further research, particularly involving hybrid approaches

  17. References • A. Bar-Noy, D. Moody, A Tiling Approach for Fast Implementation of the Travelling Tournament Problem, PATAT (2006) 351-358. • K. Easton, G. Nemhauser, M. Trick, Solving the Travelling Tournament Problem: A Combined Integer Programming and Constraint Programming Approach, PATAT (2002) 100-109. • M. Henz, T. Muller, S. Thiel, Global Constraints for Round Robin Tournament Scheduling, European Journal of Operational Research 153 (2004) 92-101. • R. V. Rasmussen, M. A. Trick, Round Robin Scheduling – A Survey, European Journal of Operational Research 188 (2008) 617-636.