Recep Arslan Zeynep Gökçe Balıkçıoğlu Melik Kaya

1. Optimization Process to Schedule the First Division of Argentine’s Professional Volleyball League Recep Arslan Zeynep Gökçe Balıkçıoğlu Melik Kaya

2. Introduction

3. General Information About the League • Teams are grouped into couples with respect to geopgraphic locations. • Matches are held on thursdays and saturdays • Each consecutive thursdays and saturdays, each couple plays against another couple.

4. Main objective of sports league is to minimize cost or travel distances. • Application areas of OR in sports are soccer, basketball, hockey, cricket.

5. Why important to minimize travel distances? • Home locations are scattered around the country • Teams do not go back home between away matches (1) how to couple teams (2) how to schedule matches

6. What is TTP(travelling tournament problem)? • Scheduling a double-round robin tournament with minimum cost or distance. • Single-round robin tournament • Double-round robin tournament

7. Computational Developments in TTP • Benders Approach • Easton & Irnich Approach: Branch and price procedure • DFS (depth first search) application

8. The scheduling problem is a practical application of TTP. • The schedules are obtained by integer programming and tabu search techniques. • This case is the first application of the TTP in the literature.

9. The League

10. GeneralInformation 11 teams in 2008-2009, 2009-2010; 12 teams in 07-08, 10-11 Generally Thursday & Saturday matches (called as weekend) Considerable long distances between venues Long travels with bus Increasing attention in recent years Successful national teams Increasing numbers of sponsorships Increasing competition

11. Top-8 first playoffs Top-4 semifinals Regular Season General Information FINALS Best-of-five tournament Double round-robin Best-of-seven tournament Double round-robin: Classical format with several leagues Best-of-five tournament: Three wins required Best-of-seven tournament: Four wins required

12. Coupling Geographically closed teams are coupled A team & B team in a couple Every weekend Thursday: Ai vs Bj & Ajvs Bi Saturday: Ai vs Bi & Aj vs Bj In the case n=11; uncoupled team vs a couple – every couple member plays one match in this weekend (bye) What about the matches within a couple?

13. Intracouple Weekends Two teams in the same couple plays w/ each other If n= 12, six matches uniformly distributed in Thursday and Saturday If n=11, five matches. Uncoupled team has a bye. In 2007-2008 & 08-09 Weeks 1 & 7 are intracouple, mirrored schedule In 09-10 Weeks 6 & 12, nonmirrored schedule In 10-11 Weeks 1 & 12, mirrored schedule Mirrored Schedule: If schedule of 1st half = schedule of 2nd half

14. Properties of Coupled Format It reduces manual and computational burden If no couples, n-teams schedule If couples, (n/2)-teams schedule It introduces a simple but effective fair-competition At most 3 strong teams in league, not in the same couple No team will have to play against two strong teams on same weekend Provides good management to travel distances Triangular inequality Reduced fatigue of the players

15. Objectives « No team can play more than two consecutive home or away weekends » Minimize the total travel distances From 09-10, travel equity is also considered As a result, problem is a special case of TTP, Couples replacing teams Weekends replacing matches L=1, U=2.

16. The Scheduling Process

17. Introduction • Needs two key decision: • How to couple the teams? • How to schedule matches between the team couples • Two-stage process: • Designing the team couples • Scheduling the matches with related to Stage 1

18. Stage 1: Designing Team Couples • Up to 2008-2009, manual coupling • Simplest approach: determine the minimum-weight matching • Only considers the travel distances between the teams in same couple • Fails to consider the distances between different couples • Optimal-tour coupling problem (discussed later) • Desired constraints • To avoid long THU-SAT trips for visiting teams • To ensure that strong teams are not in the same couple • To incorporate spectator attendance constraints

19. Stage 2: Scheduling the Matches • Determined couples, TTP with n=6. • Two possibilities about tours • Home & Away status of intracouple weeks are not included in model, arranged manually

20. Stage 2: Requirements • Each couple must play one of weekends 2 & 3 at home and the other away. - applied in 07-08, intracouples are 1&7. -Ensures no team played three consecutive weekends away. • Certain teams cannot use their stadiums on prespecified weekends. - Related with other local sport organizations • The matches on a prespecified weekend must be played near a specific city. - Super 8, two-venues organization, starts at Tuesday. Thanks to such constraints, solution times became shorter.

21. Stage 2: Objective Functions • In 2007-2008, 08-09: Minimize total travel distance • In 08-09, distances > 1080 km were double penalized. • Minimize the distance of most-traveled team • Only considers outlying teams, inefficient for central teams • Longer total travel distances, not used further • Minimize the gap between most-traveled and least-traveled • Similar results with previous one, not used further • In 09-10, 10-11: Minimize a combination of total travel distance and distance of most traveled

22. Stage 2: Scheduling the Matches • Both mirrored and nonmirrored schedule requests • Integer Programming and Tabu Search Algorithm • Tabu Search is adapted for TTP by Cardemil and Duran (2004) • Mirrored version is solvable in one to six minutes by CPLEX • Nonmirrored version needs tabu search • Tabu search is 3% better than in one minute with comparison to CPLEX solution obtained in 10 hours

23. Tour Coupling Model:

24. Integer Programming Model for Scheduling Matches • C = {1,2,…,[n/2]} set of couples • W = {1,2,…,2*[n/2]} set of weekends For i & j C; k W For i C; k W For i,j,h C; k W U {0}; we introduce binary trip variable as: • zijhk = 1, only if couple i plays away at couple j on weekend k and then away at couple h on weekend k+1.

25. IP Model for Scheduling Matches Cont’d • ziiik = 1, if couple i stays at home on weekends k and k+1. Assumption: Each team returns home before and after each intracouple match. • For i C, iA P & iB P with iA iB and iA is the A team of couple i and iB is the B team of couple i. • Intracouple weekends are k1 W and k2 W Assumption: k1 + 1 < k2 (these weekends are not consecutive) • For i C and k W, the distance traveled by iA and the distance traveled by iB on the weekend k is given by:

27. The Model is:

28. Model Cont’d

29. The TABU Search Heuristic • They have used following neighborhoods throughout the search process which gives different amounts of total travel distances. • Start with an empty schedule & randomly add the matches. • There are 4 exchanges to reach best solution • Partial-Weekend Exchange W1 W2 W1 W2 SWAP C1 & C3 C1-C3 C2-C4 C1-C4 C2-C3 C1-C4 C2-C3 C1-C3 C2-C4

30. W2 W1 • 2) Weekend Exchange Saturday E-C G-A I-D K-B F-L H-J Thursday A-B C-D E-F G-H I-J K-L Saturday A-D C-B E-H G-F I-L K-J Thursday E-A G-C I-B K-D F-J H-L SWAP

31. 3) Couple Exchange • A & C vs. B & D Change the couples: A & B vs. C & D • 4) Home-away Exchange Thursday B-A D-C F-E H-G J-I L-K Saturday D-A B-C H-E F-G L-I J-K

32. Results

33. Comparison with 2005-2006 and 06-07 schedules, global savings of $60.000

34. Discussion: What about Turkish Leauges?

35. Turkish Men’s Volleyball League • No TTP effort on professional volleyball leagues • Seems to form six feasible couples • Relatively long distances – From İzmir to Gümüşhane: 1.375 km ~ 15 hours by bus

36. User’s Opinion

37. Leonardo Carod – ACLAV president « The results were very much appreciated, especially considering the rapid solutions and the various proposals submitted for our analysis. We are extremely satisfied and expect to continue using this mathematical system in cooperation with the University and their research team.»

38. Future Plans

39. Future Plans Enhance accommodation costs and travel distances to optimize total costs.