Scheduling the German Basketball League

Transcription

1 Scheduling the German Basketball League Stephan Westphal Institute for Numerical and Applied Mathematics, University of Göttingen, Germany Daniel H. Wagner Prize for Excellence in Operations Research Practice, October 7, 2013

2 The German Basketball League Founded 1964 First season 1966 : 2 leagues with 10 teams each Since 2003/2004 : First League with 16 teams Since 2006/2007 : First League with 18 teams

3 The German Basketball League Basketball is gaining popularity in Germany. Season Spectators Average Average Capacity Average Utilization 2006/ % 2007/ % 2008/ % 2009/ % 2010/ % 2011/ %

4 Sports Scheduling Instance: n Teams, n1 time slots,n*(n1)/2 matches Task: A schedule for the matches between these teams (SingleRoundRobinTournament) Constraints: Every encounter exactly once Exactly one match for every team in each time slot

5 Sports Scheduling Is it really hard to find a schedule for a sports league? Day 1: Day 2: Day 3: Problem: The other games cannot be assigned to the remaining time slots!!

6 The German Basketball League Task: Find a schedule that fulfills the following requirements in a best possible way: Optimal compatibility to availabilty of sports sites Minimization of successive homegames (home breaks) Minimization of successive awaygames (away breaks) Utilization by the media Derby matches shall be assigned to specific day Additional wishes for home/away games or specific encounters on single days.

7 HomeAvailabilityConstraints German soccer stadiums are always accessible for soccer matches. Any german soccer team can always play in its stadium.

8 HomeAvailabilityConstraints German basketball teams play in multipurpose arenas.

9 HomeAvailabilityConstraints Example: Lokhalle in Göttingen

10 HomeAvailabilityConstraints German basketball teams play in multipurpose arenas. SportsSites are not always available. This is the most important Problem for the German Basketball League. Postponed games Less ticket sales Unbalanced standings

11 Minimization of successive homegames Homegames should be followed by Away games and vice versa Less tickets might be sold otherwise.

12 Utilization by the media The German TV Channel Sport 1 offers 30 broadcasting slots for season 2011/2012 Optimal Distribution of top games over these slots is mandatory

13 Minimization of distances on derby day On January 2 the sum of the distances Driven by the teams (and the fans) shall be minimized.

14 First Approach: The Canonical Schedule

15 First Approach: The Canonical Schedule A heuristical approach Day 1: Day 2: Day 3: Day 4: Day 5: We call this schedule the Canonical schedule

16 First Approach: The Canonical Schedule 1. Generate Canonical Schedule First half of season 1. Repeat games of first half with changed homefieldadvantage Second half of season

17 First Approach: The Canonical Schedule Observation: The whole schedule just depends on the assignment of the first day. Find best assignment of teams to nodes of graph representing the first day. => n! different schedules

18 First Approach: The Canonical Schedule Result: Viol. Availability Constraints: 31 Breaks: 48 Viol. HomeAwayWishes: 3 Viol. Encounters: 2 Distance driven on derby day : 3462 km Computation Time : 14.9 s

19 First Approach: The Canonical Schedule + Encounter Optimization Result: Viol. Availability Constraints: 31 Breaks: 48 Viol. HomeAwayWishes: 3 Viol. Encounters: 3 Distance driven on derby day : 2288 km Computation Time : 30.3 s

20 First Approach: The Canonical Schedule +Encounter Optimization+Break Minimization Result: Viol. Availability Constraints: 22 Breaks: 120 Viol. HomeAwayWishes: 4 Viol. Encounters: 3 Distance driven on derby day : 2288 km Computation Time : 75.5 s

21 Second Approach:VenueBased Optimization

22 Second Approach:VenueBased Optimization Result: Viol. Availability Constraints: 14 Breaks: 182 Viol. HomeAwayWishes: 5 Viol. Encounters: 2 Distance driven on derby day : 1097 km Computation Time : s

23 Time to try something new

24 Third Approach: Asymmetric Scheduling

25 Third Approach: Asymmetric Scheduling Every symmetric plan violates at least 14 homeavailability constraints. What about asymmetric plans? We still need: Every pair of teams meets exactly once per half. Each team plays 89 home games in each half. It s time for an ILP

26 Third Approach: Asymmetric Scheduling T:={1,2,,n} : set of teams, D:={1,2,,2n2} : set of time slots. Decision Variables 1, if i plays j at time d at i s site x i,j,d = 0, otherwise w i,d = 1, if i is having a break at time d 0, otherwise

27 Third Approach: Asymmetric Scheduling Objective Function: (HomeAvailability) (BreakMinimization) (TVBroadcasting) (Derbys)

28 Third Approach: Asymmetric Scheduling Constraints: (One Game per time) (Every match once) (Once in 1st half) (Once in 2nd half) (Derby)

29 Third Approach: Asymmetric Scheduling Constraints: (HomeBreaks) (AwayBreaks) (Broadcasting slots 1) (Broadcasting slots 2) (Broadcasting slots 3)

30 Third Approach: Asymmetric Scheduling Result: Viol. Availability Constraints: 0 Breaks: 146 Viol. HomeAwayWishes: Viol. Encounters: Distance driven on derby day : 1653 km Computation Time : 5613 s (Interruption after 12h)

31 Third Approach: Asymmetric Scheduling 146 breaks: Not acceptable

32 Third Approach: Asymmetric Scheduling Conclusions: Easy to formulate Not applicable to NonCommercial Solver GLPK anymore. With CPLEX or Gurobi: Solver gets stuck in suboptimal branch Runs forever Solution not good enough

33 Fourth Approach: The Hybrid Model

34 Fourth Approach: The Hybrid Model Approach: 1. Compute startsolution S with heuristic model. (VenueBased) 2. Use S as warmstart solution for asymmetric IPModel. 3. Fix some variables. 4. Several more runs of IP, thereby relaxing constraints, fixing variables, Postoptimize HAAssigment

35 Fourth Approach: The Hybrid Model Result: Viol. Availability Constraints: 0 Breaks: 74 Viol. HomeAwayWishes: 1 Viol. Encounters: 0 Distance driven on derby day : 1097 km Computation Time : 1055 s

36 Evaluation

37 From Model to Scheduling Solution A standalone software has to be easily operable by the project partner GUI is mandatory Interaction with solver necessary capable to solve the problems of the next seasons as well very general formulation of constraints robust and adjustable behaviour of optimizer

38 The Beko BBL Spielplanoptimierer 1.0 Entry of data Solution Scenario Handling Beko BBL League Optimizer

39 The Beko BBL Spielplanoptimierer 1.0 Entry of data for teams: name, attractivity for TV broadcasting, GPScoordinates (for derbies) Beko BBL League Optimizer

40 The Beko BBL Spielplanoptimierer 1.0 Management of rounds Beko BBL League Optimizer

41 The Beko BBL Spielplanoptimierer 1.0 Sports site availability 1= site available 0= sits already blocked Beko BBL League Optimizer

42 The Beko BBL Spielplanoptimierer 1.0 Administration of HomeAway Wishes Beko BBL League Optimizer

43 The Beko BBL Spielplanoptimierer 1.0 Administration of Encounter Wishes Beko BBL League Optimizer

44 The Beko BBL Spielplanoptimierer 1.0 Priorities for optimization, determination of derby day Beko BBL League Optimizer

45 The Beko BBL Spielplanoptimierer 1.0 Beko BBL League Optimizer Display of generated schedule Various Filters can be used to show the result of the optimization.

46 The Beko BBL Spielplanoptimierer 1.0 Display of solution quality: Availability of sports sites Distribution of Wednesdays Distances on derby day Beko BBL League Optimizer

47 The Beko BBL Spielplanoptimierer 1.0 Display of solution quality: HomeAwayPatterns Distribution of top games for TVbroadcasting Beko BBL League Optimizer

48 The Beko BBL Spielplanoptimierer 1.0 Display of progress of optimization and control of optimization process Beko BBL League Optimizer

49 The Beko BBL Spielplanoptimierer 1.0 Comparison of solutions Beko BBL League Optimizer

50 Thank you very much!

51 Evaluation

52 Sports Scheduling Instance: n Teams, n1 time slots,n*(n1)/2 matches Task: A schedule for the matches between these teams (SingleRoundRobinTournament) Constraints: Every encounter exactly once Exactly one match for every team in each time slot

53 Sports Scheduling Is it really hard to find a schedule for a sports league? Day 1: Day 2: Day 3: Problem: The other games cannot be assigned to the remaining time slots!!

54 Sports Scheduling The first n/2+2 time slots can always be chosen in a way such that the given assignment cannot be extended to a feasible schedule for all n 10. [Rosa, Wallis, 1982] Any assignment for the first three days can be extended to a feasible schedule for all n 8. [Rosa, Wallis, 1982]

55 Sports Scheduling A heuristical approach Day 1: Day 2: Day 3: Day 4: Day 5: We call this schedule the Canonical schedule

56 Double RoundRobinTournament Instance: Task: n teams, 2*(n1) time slots, n*(n1) matches Assign matches to time slots Constraints: Every team has to play against each other team once at his home venue (home game) and once at the other team s venue (away game). Every team plays exactly one match per time slot Double RoundRobin Tournaments can be derived from Single RoundRobin Tournaments by Mirroring.

57 HomeAvailabilityConstraints Lemma: There is always a schedule which obeys half of the Home AvailabilityConstraints. Proof: Consider any schedule S and the schedule S which is derived from S by changing all of the home field advantages. As each HomeAvailabilityConstraints is satisfied by either S or S, the claim follows.

58 Breaks After each home game there should be an away game and after each away game there should be a home game. A violation of this rule is called a break. Observation: On each day, the number of home and away breaks is the same. Proof: Suppose not. HAHA HAHA AHAH AHAH A A H A

59 Breaks Lemma Every SingleRoundRobinTournament with n teams has at least n2 breaks. Proof Suppose not. Then, there are 3 teams a,b,c which do not have a break at all a: AHAHAHAHAH b: AHAHAHAHAH c: HAHAHAHAHA a can never play against b.

60 Breaks Lemma Every SingleRoundRobinTournament with n teams has at least n2 breaks. Corollary Every DoubleRoundRobinTournament with n teams has at least 2n4 breaks.

61 Sports Scheduling Instance: n Teams, n1 time slots,n*(n1)/2 matches Task: A schedule for the matches between these teams (SingleRoundRobinTournament) Constraints: Every encounter exactly once Exactly one match for every team in each time slot

62 Sports Scheduling Is it really hard to find a schedule for a sports league? Day 1: Day 2: Day 3: Problem: The other games cannot be assigned to the remaining time slots!!

63 Sports Scheduling The first n/2+2 time slots can always be chosen in a way such that the given assignment cannot be extended to a feasible schedule for all n 10. [Rosa, Wallis, 1982] Any assignment for the first three days can be extended to a feasible schedule for all n 8. [Rosa, Wallis, 1982]

64 Second Approach:VenueBased Optimization Every plan we consider is symmetric in the following way: Match AB on day d implies match BA on day d+17 (d =1,...,17). If A cannot play at home on day d and d+17, there will be a violation in any schedule.

65 Second Approach:VenueBased Optimization HomeAvail. (d) HomeAvail. (d+17) Team A True True Team B False True 0 Violation HomeAvail. (d) HomeAvail. (d+17) Team A False True Team B False False 1 Violation HomeAvail. (d) HomeAvail. (d+17) Team A False False Team B False False 2 Violations

66 Second Approach:VenueBased Optimization Approach: 1. Assign games to days without Home /AwayAssignment such that number of violations is minimized (Formulation as IP) Schedule S 2. Optimize HomeAwayAssignment (IPFormulation)

67 Sports Scheduling The first n/2+2 time slots can always be chosen in a way such that the given assignment cannot be extended to a feasible schedule for all n 10. [Rosa, Wallis, 1982] Any assignment for the first three days can be extended to a feasible schedule for all n 8. [Rosa, Wallis, 1982]

68 The German Basketball League Basketball is gaining popularity in Germany. Season Spectators Average Average Capacity 2004/ / ,50% 2006/ ,80% 2007/ ,10 % 2008/ ,00 % 2009/ ,12 %

69 Comparison of the models