SCHEDULING THE BELGIAN FOOTBALL LEAGUE DURING SPECIAL PERIODS
Word count: 27.411
Sam Wauters Student number: 01306574
Promotor / Supervisor: Prof. Dr. Dries Goossens Commissioner: Xiajie Yi
Master’s Dissertation submitted to obtain the degree of:
Master in Business Engineering: Operations Management
Academic year: 2018-2019
Deze pagina is niet beschikbaar omdat ze persoonsgegevens bevat. Universiteitsbibliotheek Gent, 2021.
This page is not available because it contains personal information. Ghent University, Library, 2021.
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Acknowledgements
I would like to thank my supervisor, professor Goossens for giving me the opportunity to do this research under his guidance. Next, I would like to thank Xiajie Yi for the continuous and helpful feedback during this research process.
A special thanks goes to my parents and sister for supporting me in my studies throughout the years by giving encouragements during the exams and the writing of this thesis. I also want to thank my girlfriend Sarah for the countless hours we worked together to write our dissertations.
This accomplishment would not have been possible without the support of each of the above.
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Table of Content
List of Tables & Figures______X
1. Introduction ______1
Part 1: Literature ______3
2. Round-Robin Scheduling ______5 2.1. Introduction ______5 2.2. Terminology ______6 2.3. Objectives ______10 2.3.1. Break Minimization ______10 2.3.2. Travelling Distance Minimization ______10 2.4. Constraints ______14 2.4.1. Basic Constraints ______14 2.4.2. Externally given constraints ______14 2.4.3. Fairness Constraints ______16 2.5. Solution Methods ______18 2.5.1. Decomposition ______18 2.5.2. Integer Programming ______19 2.5.3. Constraint programming ______26 2.5.4. Metaheuristic ______26
3. Belgian Pro League Football ______28 3.1. Introduction & league format ______28 3.2. Stakeholders ______29 3.2.1. The Belgian Pro League ______29 3.2.2. The TV broadcasters ______30 3.2.3. The police and the taxpayer ______31 3.2.4. The clubs and their fans ______31 3.3. Scheduling Methods ______32
Part 2: Case Study: Distance Minimization in the Jupiler Pro League ______35
4. Problem Description ______37 4.1. Introduction ______37 4.2. Solution Method ______38
VII
4.3. Constraints ______39 4.4. Data ______41 4.4.1. Travelling Distance Matrix ______41 4.4.2. Actual Schedules & Distances ______42 4.5. Special Period Fairness Approaches ______44 4.5.1. Fairness Approach 1: Total Distance Minimization ______44 4.5.2. Fairness Approach 2: Limited Travelling Distance per Team per Round ______44 4.5.3. Fairness Approach 3: Limited Difference Between Teams ______44 4.5.4. Fairness Approach 4: Limited Travelling Distance per Team over Special Rounds ____ 45
5. Model ______46 5.1. Basic Model ______46 5.1.1. Indices ______46 5.1.2. Decision Variables ______48 5.1.3. Parameters ______48 5.1.4. Objective Function ______49 5.1.5. Constraints ______50 5.2. Fairness Approaches Constraints ______52 5.2.1. Fairness Approach 1: Total Minimization ______52 5.2.2. Fairness Approach 2: Limited Travelling Distance per Team per Round ______52 5.2.3. Fairness Approach 3: Limited Difference Between Teams ______53 5.2.4. Fairness Approach 4: Limited Travelling Distance per Team over Special Rounds ____ 53
6. Simulation & Results ______54 6.1. Fairness Approach 1: Total Distance Minimization ______54 6.1.1. 2-Round Special Period ______55 6.1.2. 4-Round Special Period ______56 6.2. Fairness Approach 2: Limited Travelling Distance per Team per Round ______58 6.2.1. 2-Round Special Period ______59 6.2.2. 4-Round Special Period ______61 6.3. Fairness Approach 3: Limited Difference Between Teams ______63 6.3.1. 2-Round Special Period ______64 6.3.2. 4-Round Special Period ______65 6.4. Fairness Approach 4: Limited Travelling Distance per Team over Special Rounds ______67 6.5. Additional Approaches: ______70 6.5.1. Additional Approach 1: Using the actual home-away patterns ______70 6.5.2. Additional Approach 2: Combining Approach 2 and 4 ______74
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7. Discussion ______77 7.1. Season 2018-2019 ______77 7.2. Season 2017-2018 ______80 7.3. Season 2016-2017 ______81
8. Conclusion______83
References ______85
Appendices ______91
A. Distances ______93
B. Actual Special Period Schedules ______96
C. Attractive Matches ______97
D. CPLEX Model & Data ______98
E. Resulting Tournament Schedules: Fairness Approach 1 ______106
F. Resulting Tournament Schedules: Fairness Approach 2 ______109
G. Resulting Tournament Schedules: Fairness Approach 3 ______112
H. Resulting Tournament Schedules: Fairness Approach 4 ______115
I. Resulting Tournament Schedules: Additional Approach 1 ______117
J. Resulting Tournament Schedules: Additional Approach 2 ______120
K. Home-Away Patterns ______122
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List of Tables & Figures
Table 2-1: Timetable ...... 7 Table 2-2: Complementary pattern sets for a 6-team tournament ...... 7 Table 2-3: Mirrored double round-robin tournament ...... 9
Table 3-1: Basic Match Schedule...... 32
Table 4-1: Travelling Distances per Team: Season 2018-2019 ...... 42 Table 4-2: Travelling Distances per Team: Season 2017-2018 ...... 43 Table 4-3: Travelling Distances per Team: Season 2016-2017 ...... 43
Table 6-1: Total Overall Distances & Comparison Approach 1 ...... 54 Table 6-2: 2-Round Special Period Schedule 2018-2019 (Fairness 1) ...... 55 Table 6-3: 2-Round Individual Distance Comparison 2018-2019 (Fairness 1) ...... 55 Table 6-4: 2-Round Special Period Schedule 2017-2018 (Fairness 1) ...... 55 Table 6-5: 2-Round Individual Distance Comparison 2017-2018 (Fairness 1) ...... 55 Table 6-6: 2-Round Special Period Schedule 2016-2017 (Fairness 1) ...... 56 Table 6-7: 2-Round Individual Distance Comparison 2016-2017 (Fairness 1) ...... 56 Table 6-8: 4-Round Special Period Schedule 2018-2019 (Fairness 1) ...... 56 Table 6-9: 4-Round Individual Distance Comparison 2018-2019 (Fairness 1) ...... 56 Table 6-10: 4-Round Special Period Schedule 2017-2018 (Fairness 1) ...... 57 Table 6-11: 4-Round Individual Distance Comparison 2017-2018 (Fairness 1) ...... 57 Table 6-12: 4-Round Special Period Schedule 2016-2017 (Fairness 1) ...... 57 Table 6-13: 4-Round Individual Distance Comparison 2016-2017 (Fairness 1) ...... 57 Table 6-14: Total Overall Distances & Comparison Approach 2 ...... 58 Table 6-15: 2-Round Special Period Schedule 2018-2019 (Fairness 2) ...... 59 Table 6-16: 2-Round Individual Distance Comparison 2018-2019 (Fairness 2) ...... 59 Table 6-17: 2-Round Special Period Schedule 2017-2018 (Fairness 2) ...... 60 Table 6-18: 2-Round Individual Distance Comparison 2017-2018 (Fairness 2) ...... 60 Table 6-19: 2-Round Special Period Schedule 2016-2017 (Fairness 2) ...... 60 Table 6-20: 2-Round Individual Distance Comparison 2016-2017 (Fairness 2) ...... 60
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Table 6-21: 4-Round Special Period Schedule 2018-2019 (Fairness 2) ...... 61 Table 6-22: 4-Round Individual Distance Comparison 2018-2019 (Fairness 2) ...... 61 Table 6-23: 4-Round Individual Distance Comparison 2017-2018 (Fairness 2) ...... 62 Table 6-24: 4-Round Special Period Schedule 2017-2018 (Fairness 2) ...... 62 Table 6-25: 4-Round Special Period Schedule 2016-2017 (Fairness 2) ...... 62 Table 6-26: 4-Round Individual Distance Comparison 2016-2017 (Fairness 2) ...... 62 Table 6-27: Total Overall Distances & Comparison Approach 3 ...... 63 Table 6-28: 2-Round Special Period Schedule 2018-2019 (Fairness 3) ...... 64 Table 6-29: 2-Round Individual Distance Comparison 2018-2019 (Fairness 3) ...... 64 Table 6-30: 2-Round Special Period Schedule 2017-2018 (Fairness 3) ...... 64 Table 6-31: 2-Round Individual Distance Comparison 2017-2018 (Fairness 3) ...... 64 Table 6-32: 2-Round Special Period Schedule 2016-2017 (Fairness 3) ...... 65 Table 6-33: 2-Round Individual Distance Comparison 2016-2017 (Fairness 3) ...... 65 Table 6-34: 4-Round Special Period Schedule 2018-2019 (Fairness 3) ...... 65 Table 6-35: 4-Round Individual Distance Comparison 2018-2019 (Fairness 3) ...... 65 Table 6-36: 4-Round Special Period Schedule 2017-2018 (Fairness 3) ...... 66 Table 6-37: 4-Round Individual Distance Comparison 2017-2018 (Fairness 3) ...... 66 Table 6-38: Total Overall Distances & Comparison: Approach 4 ...... 67 Table 6-39: 4-Round Individual Distance Comparison 2018-2019 (Fairness 4) ...... 68 Table 6-40: 4-Round Special Period Schedule 2018-2019 (Fairness 4) ...... 68 Table 6-41: 4-Round Individual Distance Comparison 2017-2018 (Fairness 4) ...... 68 Table 6-42: 4-Round Special Period Schedule 2017-2018 (Fairness 4) ...... 68 Table 6-43: 4-Round Individual Distance Comparison 2016-2017 (Fairness 4) ...... 69 Table 6-44: 4-Round Special Period Schedule 2016-2017 (Fairness 4) ...... 69 Table 6-45: Total Overall Distances & Comparison: Additional Approach 1 ...... 70 Table 6-46: 2-Round Special Period Schedule 2018-2019 (Additional Approach 1) ...... 71 Table 6-47: 2-Round Individual Distance Comparison 2018-2019 (Additional Approach 1) ...... 71 Table 6-48: 2-Round Special Period Schedule 2017-2018 (Additional Approach 1) ...... 72 Table 6-49: 2-Round Individual Distance Comparison 2017-2018 (Additional Approach 1) ...... 72 Table 6-50: 2-Round Special Period Schedule 2016-2017 (Additional Approach 1) ...... 72 Table 6-51: 2-Round Individual Distance Comparison 2016-2017 (Additional Approach 1) ...... 72 Table 6-52: 4-Round Special Period Schedule 2018-2019 (Additional Approach 1) ...... 73 Table 6-53: 4-Round Individual Distance Comparison 2018-2019 (Additional Approach 1) ...... 73 Table 6-54: 4-Round Special Period Schedule 2017-2018 (Additional Approach 1) ...... 73
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Table 6-55: 4-Round Individual Distance Comparison 2017-2018 (Additional Approach 1) ...... 73 Table 6-56: 4-Round Special Period Schedule 2016-2017 (Additional Approach 1) ...... 74 Table 6-57: 4-Round Individual Distance Comparison 2016-2017 (Additional Approach 1) ...... 74 Table 6-58: Total Overall Distances & Comparison: Additional Approach 2 ...... 74 Table 6-59: 4-Round Special Period Schedule 2018-2019 (Additional Approach 2) ...... 75 Table 6-60: 4-Round Individual Distance Comparison 2018-2019 (Additional Approach 2) ...... 75 Table 6-61: 4-Round Special Period Schedule 2017-2018 (Additional Approach 2) ...... 75 Table 6-62: 4-Round Individual Distance Comparison 2017-2018 (Additional Approach 2) ...... 75 Table 6-63: 4-Round Special Period Schedule 2016-2017 (Additional Approach 2) ...... 76 Table 6-64: 4-Round Individual Distance Comparison 2016-2017 (Additional Approach 2) ...... 76
Table 7-1: 2-Round Special Period Results Comparison: Season 2018-2019 ...... 77 Table 7-2: 4-Round Special Period Results Comparison: Season 2018-2019 ...... 78 Table 7-3: 2-Round Special Period Results Comparison: Season 2017-2018 ...... 80 Table 7-4: 4-Round Special Period Results Comparison: Season 2017-2018 ...... 80 Table 7-5: 2-Round Special Period Results Comparison: Season 2016-2017 ...... 81 Table 7-6: 4-Round Special Period Results Comparison: Season 2016-2017 ...... 81
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1. Introduction
Every sports competition has the need for a suitable schedule that allows a fair and practically feasible tournament. The amount of money in professional sports has increased substantially in the past decades. For example, according to a research by Deloitte (2018) the European football market is worth €25.5 billion. Since it is such a big business, the different stakeholders demand a fair and suitable tournament to assure their revenues and wishes. Also, there is a wide application for scheduling problems and methods, not only in professional but also recreational sports leagues (Russel & Urban, 2006). It seems easy to set-up a tournament, but the rise in requirements and constraints has made practical problems difficult to optimize. The increase in computational capabilities of microprocessors and development of more efficient algorithms allowed researchers to deal with these complex problems. Because of this, sports scheduling has become an important research area within operations research (Nurmi, et al., 2010).
The scheduling of the Belgian first division of football is one of these practical problems that is being solved by using methods originating from the field of operations research. Continuous changes to this double round-robin tournament and the requirements of the stakeholders make this scheduling problem an almost unending task with constant room for improvement. The objective of this research is further improving the tournament schedule for the Belgian first division, more specifically by trying to reduce the travelling distance for the teams during special periods. These include the period around Christmas and other days with potential bad weather during winter. It can be assumed that the teams and fans are not keen on travelling a long distance to see their team play during these special periods. By building upon the research that has already been done for scheduling the Belgian football and other tournament problems, a suitable method to solve this distance minimization problem will be set-up.
The remainder of this paper is structured as follows. The first part consists of the literature research. The terminology that is used in round-robin scheduling literature is explained, together with practical objectives, requirements and solution methods used in practice. The literature concerning the characteristics and scheduling of the Belgian football is also discussed. The second part contains the actual research for the reduction of travelling distance during the special periods. The problem is explained more in depth, followed by a description of the actual model used and the results. These results are then discussed and summarized in the conclusion. Some limitations of this research and recommendations for future research are also given in the conclusion.
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Part 1: Literature
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2. Round-Robin Scheduling
2.1. Introduction De Werra (1980) was one of the earliest researchers that used mathematical methods to schedule sports leagues. He utilized concepts from graph theory to deal with geographical constraints within a tournament. Since then, many studies and papers have been written on round-robin scheduling. A listing of sports scheduling principles, methodologies and applications can be found in the bibliography by Kendall, Knust, Ribeiro, & Urrutia (2010). Many of the studies in the bibliography build on the concepts proposed by de Werra and introduce several other specific terms to describe the scheduling problems and their solutions. First, a basic understanding of these terms is necessary to better understand the round-robin scheduling concepts and its practical applications.
Various goals exist when setting up a tournament schedule. Explicit objectives like distance and cost minimization are possible or the goal may simply be to set-up a feasible schedule. The two most common objectives will be explained more in depth, particularly distance minimization.
When setting up a schedule for sports leagues, many wishes and constraints have to be taken into account. The various stakeholders (teams, fans, local governments, TV broadcasters, etc.) often have conflicting requirements that they want to see represented in the schedule. The constraints that are most commonly used in sports scheduling literature are explained more in detail.
The various methods used today to solve round-robin scheduling problems are explained next. Ranging from combinatorial approaches, integer programming (IP), constraint programming (CP) to metaheuristic techniques. Each has its own characteristics, strengths, weaknesses and the possibility to find an optimal or near-optimal solution.
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2.2. Terminology
Round-Robin Tournament A tournament where all teams meet all other teams a defined number of times is called a round- robin tournament (Rasmussen & Trick, Round robin scheduling – a survey, 2008).
The most common format is the double round-robin tournament, where teams have two games against each other. The three biggest European football leagues (England, Spain and Germany) (Deloitte, 2018) all use the double round-robin format. The number of games that each team must play in this case, is given by the number of teams in the tournament. In a tournament with n teams, each team will play 2(n - 1) games (Carlsson, Johansson, & Larson, 2017). For example, each of the 20 teams in the English Premier League will play 38 games.
Triple and quadruple round-robin tournaments also occur in practice, with three and four matches between the teams respectively. For example, from 1995 till 2016, the Danish top football league was played in a triple round-robin tournament by 12 teams. Each team had to play 33 games (Rasmussen, 2008).
The total number of games to be played in a k round-robin tournament is equal to (n/2)(n-1)k, with n the number of teams (Drexl & Knust, 2007). So, in a double round-robin tournament with 16 teams, a total of 240 games must be scheduled. For the Danish football league, the number of games was equal to 198.
Time slots/rounds, bye, temporally constrained & relaxed and timetables Games between two teams must be assigned to time slots so that each team has at most one game in each time slot or round (Rasmussen & Trick, 2008), with games in one round usually played in the same midweek or weekend. With an odd number of teams, every round one team has a bye which means they have no game scheduled. In this case a dummy team can be added to make the scheduling process easier. For each round, the team playing the dummy has a bye.
The minimum number of rounds is equal to the number of games each team must play in the entire tournament, so at least 2(n – 1) rounds are required to schedule a double round-robin tournament. When the amount of time slots equals this lower bound, the schedule is compact or temporally constrained. With an even number of teams this means each team has a match in every round. For an uneven number of teams, the necessary byes must be added.
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In a temporally relaxed schedule, the number of rounds is not minimized and generally larger than the lower bound. It is not necessary that every team plays in each round and thus they can have some rounds without a game scheduled (Drexl & Knust, 2007).
To present the allocation of games to rounds, a timetable is used. The timetable in table 2-1 shows the opponent for each team in every round of the single round-robin tournament. The rows of the timetable represent the teams and the columns represent the rounds. The value of row i and column j gives the opponent of team i on round j (Rasmussen & Trick, 2008). Rounds 1 2 3 4 5 Team 1 63524 Team 2 56413 Team 3 41652 Team 4 35261 Team 5 24136 Team 6 12345 Table 2-1: Timetable
Venue, home & away games, (complementary) patterns and balanced tournaments When two teams meet each other, one team plays at home while the other plays an away game. Each team has its own venue where its scheduled home games are played. Away games are played at opposing teams’ venues. The venues have an address that is used to determine the travelling distance between a team’s home venue and the opponent’s venue
Rounds 1 2 3 4 5 The sequence of games played at home, away or a bye scheduled for a Team 1 10101 team during the season is called the home-away pattern. Table 2-2 Team 2 01011 shows an example of a pattern set for a 6-team tournament, with 1 Team 3 10010 indicating a home game and 0 an away game. For each pair of teams Team 4 01101 (1-2, 3-4, 5-6), the patterns are complementary: if the first pattern has Team 5 10100 an away game on a certain round, the second pattern has a home game Team 6 01011 on that same round (Nurmi, et al., 2010). Table 2-2: Complementary pattern sets for a 6-team tournament
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The tournament is balanced when the deviation of number of home and away games for each team over the whole season is equal to 1. In other words, each team has an equal number of home and away games at the end of the seasons. In a double round-robin tournament, the two games between each pair of teams are required to be played in opposite venues. So, each team must play a home and an away game against every opponent. This forces the tournament to be balanced and forces the travelling distance over the whole season to be fixed. For single or multi round-robin schedules, it is often required that the schedule is balanced and that the deviation of home and away games for each pair is no more and as close as possible to one (Rasmussen & Trick, 2008). Note that the single round-robin tournament in table 2-2 is not balanced as the number of rounds is uneven.
Breaks and mirrored schedule A break occurs when a team has two consecutive home or away matches in its home-away pattern (Goossens & Spieksma, Breaks, cuts, and patterns, 2011). In table 2-2, team 2 has consecutive away games on round 3 & 4, so it’s said to have an away break on the fourth round. Team 4 has a home break on the third round.
In a single round-robin tournament, only two home-away patterns exist without breaks: starting and ending with a home (away) game. Max two teams will have a break-free pattern, since all teams must have different patterns to allow a match between each pair of teams (as two teams with an equal home-away pattern cannot play each other). This implies that the lower bound for breaks in a single round-robin schedule equals n–2. A tournament with 16 teams thus has a minimum of 14 breaks.
The minimum number of breaks in a double round-robin tournament is 2n-4. For 16 teams this means at least 28 breaks. However, schedules with this minimum number of breaks often have the undesired characteristic that the last round of the first half of the tournament has the same pairings as the first round of the second half (Drexl & Knust, 2007). In practice it is often preferred that schedule is mirrored. This means that the assignment of opponents is identical in the first and second half of the tournament for each team, except that the team playing at home becomes the away team and vice versa (Rasmussen & Trick, 2008). Table 2-3 is an example of a mirrored double round-robin tournament with 6 teams. It is proved by de Werra (1981) that the minimum number of breaks for a mirrored schedule is 3n-6. At least 42 breaks are thus needed when scheduling 16 teams.
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Rounds1 2 3 4 5 6 7 8 910 Team 1 +6 -3 +5 -2 +4 -6 +3 -5 +2 -4 Team 2 +5 -6 -4 +1 -3 -5 +6 +4 -1 +3 Team 3 -4 +1+6 -5 +2+4 -1 -6 -5 -2 Team 4 +3 -5 +2 -6 -1 -3 +5 -2 +6+1 Team 5 -2 +4 -1 +3 +6 +2 -4 +1 -3 -6 Team 6 -1 +2 -3 +4 -5 +1 -2 +3 +4 +5 Table 2-3: Mirrored double round-robin tournament
Carry-over effect In physical sports like football, with risks of injuries and fatigue, the carry-over effect might influence the performance and results of the teams. If team A is first scheduled to play against team B and then against team C in the following round, team C is said to receive a carry-over effect from team B. For example, when team B is a strong opponent with a physical style of playing, team A will be negatively affected by fatigue or possible injuries and this is an advantage for the opponent in the next round, team C. Or otherwise, if team B is relatively weak, team A could rest some players to have an advantage over its next opponent. The effect can also be psychological if team A loses against the stronger team B and consequently has a loss in morale and confidence. This will benefit team C. It is also possible that team B receives a carry-over effect from team C: if team A meets the stronger team C in the next round, they might have extra motivation for the first game against team B. It is often preferred when setting up the tournament schedule that carry-over effects are distributed over the teams in a fair way, so no team has an excessive advantage or disadvantage over the other teams (Goossens & Spieksma, 2012).
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2.3. Objectives
2.3.1. Break Minimization Breaks are generally not preferred by teams because of the impact on the results and financial reasons. By avoiding multiple rounds without a home game, regular earnings are ensured (Rasmussen & Trick, 2008). Breaks are also not preferred by the fans as they dislike long periods without a home game (Nurmi, et al., 2010). Therefore, a minimization of breaks is often an important goal when setting up a tournament schedule, especially in many professional European football leagues. As mentioned above, the lower bounds for breaks in a single and double round-robin tournament are n-2 and 2n-4, respectively. Constructive methods have been developed that are able to solve this break minimization problem. De Werra (1982) used the relationship between graph theory and tournaments to obtain feasible schedules by constructing the canonical 1- factorization method.
However as more and more requirements were considered from various stakeholders, the theoretical methods and results from graph theory were not sufficient. The addition of these constraints led to the definition of constrained minimum break problems. Since these practical problems were significantly different from each other, various methods and algorithms were developed (Rasmussen & Trick, 2008). Some of these methods will be further explained below.
2.3.2. Travelling Distance Minimization Another important objective for tournament scheduling is the minimization of the travel distances for the participating teams. The two main approaches are the Travelling Tournament Problem and the distance minimization on specific rounds during the tournament.
Travelling Tournament Problem The Travelling Tournament Problem was first presented by Easton, Nemhauser, & Trick (2001). They were inspired by previous work done for the Major League Baseball (MLB) in the United States. In the MLB, thirty teams have to play 162 games each over a season of 180 days stretching from early April till the end of September. The already difficult scheduling task is further complicated by the distances the teams have to travel. Because the teams are located all over the United States, they go on ‘road trips’, where multiple opponents are visited before they return home. The total travelling distance is thus an important factor for the teams in the MLB.
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The key to scheduling the MLB is the conflict between the travelling distances and the home-away patterns. Away trips of more than two weeks, where the teams visit 3 or 4 opponents, as well as staying at home for so long are not preferred.
This combination of travelling distance and home-away patterns is not uncommon and so the problem class of the Travelling Tournament Problem was proposed. The general Travelling Tournament Problem defined by Easton et al. (2001) gives the basic elements needed to describe a practical case. The double round-robin tournament is played by n teams. A (n x n) distance matrix D is used to represent the travelling distances between the teams’ venues. The teams begin and end the tournament at their home venue. When the teams have an away game after they are at home, they must travel from their home venue to the away venue. For consecutive away games they don’t return home, but travel between the away venues directly. As mentioned above, these consecutive away games are called road trips. Home stands occur when teams play consecutive home games. The length of the home stands or road trips is equal to the number of consecutive home or away games, respectively. A lower limit L and an upper limit U can be imposed on the length of both the home stands and road trips.
The goal is to find a schedule for the double round-robin tournament where the length of the road trips and home stands is between U and L and that the total travelling distance by all the teams is minimized. Additional constraints can be added so that the schedule is mirrored, that teams cannot play each other twice on consecutive rounds, etc.
It would seem that these problems are relatively easy to solve by combining existing break minimization methods and techniques from the Travelling Salesman Problem, which shows similarity to the distance problem. However, simultaniously tackling the optimality (with regards to the travel distance) and feasibilty issues (the home-away patterns) is very difficult and computationally complex. Several solution methods have been proposed: Easton, Nemhauser, & Trick (2003) proved the first optimal solution for an instance of 8 teams by using integer and constraint programming; Ribeiro & Urrutia (2007) used several heuristics to solve the mirrored traveling tournemant problem of the 2003 Brazilian football championship with 24 teams; Anagnostopoulos, Michel, Van Hentenryck, & Vergados (2006) proposed a simulated annealing algorithm that significantly improved results over other approaches for larger instances. An updated list of the best solutions and the details of all instances are found at Trick (2019) (https://mat.gsia.cmu.edu/TOURN/) (last updated on April 22nd 2019).
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Distance minimization on specific rounds While road trips are common in large countries like the US, they do not occur in most of the European sports leagues. The teams have only one or two games a week and they return home after every away game, generally on the same day. This implies that their total travelling distance over the whole season is fixed and so a minimization of the total distance as seen in the Travelling Tournament Problem is pointless. However, other factors can be considered to offer a more attractive schedule for the teams and their supporters. Examples are minimizing the distance travelled by all teams on midweek rounds or during selected periods or rounds.
Kendall (2008) tried to minimize the total travelling distance by all the teams in the English football leagues over the Christmas holiday period. Teams and their fans wish to limit their travel distances over this period for various reasons: avoiding trips in potential bad (snowy) weather, driving in the dark on days with less daylight, wanting to spend more time with friends and family on Christmas, etc.
He used the schedules and teams that were playing in the top four divisions in England, more specifically in the ‘Barclays Premiership’, ‘Coca-Cola Championship’, ‘Coca-Cola League One’ and ‘Coca-Cola League Two’, in the four seasons from 2002 till 2006. Each of these leagues is a stand alone double round-robin tournament, with the former consisting of 20 teams and the other three each having 24 teams. The goal was to set-up schedules for the Boxing Day (26th of December) and New Year’s Day rounds. With a total of 92 teams in all four leagues, 46 games must be scheduled in each round. An added difficulty is that a pairing system determines that various teams, for example Manchester City and Manchester United, are not allowed to have a home game on the same day. The main reason for these restrictions are police requirements to guarantee the safety. This pairing system is applied across the divisions, which means that schedules from one division cannot be set-up without taking the other divisions into account. Kendall listed some other hard constraints that could not be violated:
A team playing at home on the Boxing Day round must have an away game on the New Year’s round, and vice versa. Two teams cannot play each other on both the Boxing Day and New Year’s Day round. Teams that are paired are not allowed to play each other. The number of paired teams having a home game was not allowed to exceed given limits. These limits were based on the values from the actual used schedules.
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The number of London-based, London Premiership and Greater Manchester teams that can play at home on one round are limited.
The distances between the teams used in the model are calculated based on the postcode (ZIP code) of the teams and using a route planner (http://greenflag.co.uk).
To generate a solution a two-stage approach was used. In the first stage a Depth First Search algorithm is utilized to produce a solution for both days, for each of the four divisions. The schedules for the two rounds were a significant improvement over the actual used schedules in terms of the minimization of the travelling distance. However, the pairing constraints and London & Manchester constraints were not yet incorporated, because they work across the divisions. To incorprate these cross-divisional constraints into the schedules, the second stage with a Local Search algorithm was used.
The results were a significant reduction in travelling distance compared to the actual schedules, without violating any of the listed constraints. For the four seasons incorporated in the study, an average overall saving of 27,38% is achieved.
It should be noted that Kendall only made a schedule for the Boxing Day and New Year’s Day rounds, so it’s not sure if this distance minimization would allow the generation of a feasible schedule for the entire tournament. An extension of his research to the entire tournament with the inclusion of additional constraints is needed. He mentioned himself that he would like to do so by discussing the problem with the stakeholders and thus incorporating the issues they actually encounter.
In a more recent study, Kendall and Westphal (2013) expanded the model by going from 2 to 4 rounds being considered for distance minimization and they adressed several shortcomings in the solution method. The previously used two-phased approach was time consuming, especially with the Depth First Search phase taking up to 20 hours to complete. In addition to this, the solutions were not certain to be optimal as metaheuristics were used.
The new proposed solution method aimed to solve the problem in a single phase and considers all four divisions simultaneously. In addition to this, they suggested that the best solution with an overall minimum distance might not be accepted, but a slightly worse solution where the distance for the teams within each division is below a limited value might be preferred. This will allow for a schedule that is viewed as more fair by the teams and supporters.
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The mathematical model together with the scenarios to limit the distance are discussed in section 2.5.2.2. Solving the model in CPLEX resulted in a significant improvement over the actual schedules as well.
2.4. Constraints
2.4.1. Basic Constraints The first basic constraint that is often used in actual schedules states that each team (for n = even) must play exactly one game on each round. This makes the tournaments compact.
As mentioned above, in a round-robin tournament each team has a fixed number of matches with every other team. The second basic constraint must assure that in a k round-robin tournament, each team meets all other teams exactly k times. For a double round-robin tournament this means that every pair of two teams must have a match on 2 different rounds, one in each of the two halves of the tournament. An additional constraint states that when a match of team i against team j is first scheduled as a home game for team i, the match in the second half of the tournament must be played at team j’s venue.
These constraints are generalized by Briskorn and Drexl (2009): a k round-robin tournament with n teams has k(n-1) rounds with k matches between team i and j. At least ⌊푘/2⌋ matches of team i against team j are held at both teams’ venue.
2.4.2. Externally given constraints Venue Constraints Sometimes it is necessary that team i plays at home (or away) on a predefined round of the tournament. For towns that only have one team playing in the competition, it is preferred not to have a home game when another big event, like a festival or international conference, takes place in the vicinity of the stadium. Also, stadiums are often co-owned by the club and the city or some other public agency and they might host other events at the stadium. So, stadium availability cannot be taken for granted and must be handled by constraints when setting up the schedule (Briskorn & Drexl, 2009). These can be hard or soft constraints, depending on the reason (Schaerf, 1999).
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If, for example a venue is unavailable due to a concert or reconstruction, it can be considered a hard constraint. In case the requirement is caused by nearby events in the city, it would be a soft constraint (Rasmussen, 2008).
Region Constraints The venues of the teams are located in different regions. Some regions have more than one team and having multiple home games by these teams on the same day can cause heavy traffic and a higher demand for local police forces and firemen. A notable example is the London region where 13 professional football teams have their venue. Even with their own venue, too many home games can overload the region with traffic jams and congested public transport (Kendall, 2008). To avoid excessive problems, only a limited number of home games can be scheduled on the same day within the same region. To establish this upper bound the individual regions’ aspects like road networks and public transport must be taken into account (Briskorn & Drexl, 2009).
Attractiveness Constraints Television broadcasting stations have become one of the most important stakeholders for sports leagues. The cost of the broadcasting rights for the English Premier League Football has risen to €10.25 million (£9.2 million) (The Associated Press, 2019). To have a more optimal return on their large investments, TV broadcasters want to present as many attractive games as possible (Briskorn & Drexl, 2009). So, scheduling two attractive games at the same round might not be preferred. Instead, the broadcasters prefer an even distribution over the season or a fixed number of attractive games on a certain round. For example, in Brazil, only one match per round could be broadcast to Rio de Janeiro and São Paulo. As TV Globo wanted to generate as much revenue as possible from advertising, the goal was a maximization of number of rounds where at least one game of an elite team could be broadcast in both Rio de Janeiro and São Paulo (Ribeiro & Urrutia, 2006).
The attractiveness of a game is mostly dependent on the two teams playing each other. Based on for example the ranking in the previous seasons or popularity of the teams, the attractiveness for the supporters and consequentially the broadcasters can be identified (Briskorn & Drexl, 2009).
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Complementary constraints When two teams i and j must have a complementary home-away patters, team i must play a home game if team j is scheduled to play an away game, and vice versa (Schaerf, 1999). This occurs when two teams share the same stadium, so two home games in the same round are not possible. For example, in the Italian Football League, no more than two teams are located in each town but at most one stadium is available. These multiple complementary sets of teams are hard constraints that must be taken into account when setting up the schedule (Della Croce & Oliveri, 2006).
Sometimes there are interdependencies with other leagues. In Germany, the schedules for the ‘1. Fußballbundesliga’ and the ‘2. Fußballbundesliga’ cannot be determined independent from each other. For two teams in the same city with a team playing in the first division and the other playing in the lower division, the home-away patterns of these teams must be complementary. The schedule of the first division is determined first and then the schedule of the second division must take the resulting home-away patterns for the complementary first division teams into account (Bartsch, Drexl, & Kröger, 2006).
2.4.3. Fairness Constraints
Break constraints Since the pattern of home and away games of the teams has an important influence on the fatigue and number of visitors, there can be special requirements for these patterns in the tournament. As mentioned above, when assigning the home-away pattern a minimization of breaks can be an objective for the overall schedule. However, several other constraints concerning the breaks are often used in practice to ensure a fairer schedule for each team.
Instead of minimizing the total number of breaks, a fixed number of breaks being assigned to each team can be preferred for fairness reasons. In a single round-robin tournament for example, scheduling exactly one break for each team can be viewed as a fairer approach (Briskorn & Drexl, 2009).
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Limitations on the number of consecutive breaks or particular sequences of home and away games that should be avoided are common. For example, more than two consecutive home or away games are to be avoided, often by hard constraints. In leagues with an uneven number of teams, no more than three home (away) matches or byes in a row can be forbidden (Martin, 2002).
A beginning constraint states that each team must have a home and an away game in the first two rounds of the tournament. Similarly, an ending constraint assigns a home and away game in the last two rounds of the tournament (Rasmussen, 2008). These constraints can be summarized by stating that no team is allowed to have a break in the second and last round of the tournament. The aim of these constraints is to prevent distortions of the competition (Bartsch, Drexl, & Kröger, 2006).
Separation constraints In a non-mirrored k round-robin tournament with k≥2, a lower bound is often imposed on the number of rounds between the games of each pair of teams, to avoid that the k games are scheduled in a short time span. For example, in Austria, teams that play each other in round 17 or 18 (first half of the tournament), cannot have their second match before round 20 or 21 (second half of the tournament). In other words, at least 2 rounds must be in between the two games of each pair of teams. (Bartsch, Drexl, & Kröger, 2006). This is not relevant for mirrored schedules since there are at least n-2 rounds between the games against the same opponent.
Top team and bottom team constraints Besides the carry-over effect, other constraints concerning the strength of the teams are taken into account when scheduling tournaments. For example, in the Italian first division of football (Serie A), the top teams from the previous season are considered as seeded teams. No matches between these teams may occur in the first and last three rounds of the tournament. This is to make sure that no team has an advantage over the other teams in the beginning and in the decisive final weeks of the season (Della Croce & Oliveri, 2006). Generally, it is preferred that games against tough and weak opponents are evenly distributed over the whole tournament for each team (Briskorn & Drexl, 2009). For the German football league, the 18 teams are evenly distributed in three strength groups based on the results of the previous season. The schedule makes sure that no team must play matches against opponents from the same strength group on consecutive rounds (Bartsch, Drexl, & Kröger, 2006).
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2.5. Solution Methods
2.5.1. Decomposition Tournament scheduling problems are often solved by decomposing them into subproblems which are then solved one after the other by using various exact or heuristic algorithms. Most commonly a combination of integer and constraint programming or heuristic methods is used to solve the subproblems.
Mainly, there are two approaches that are used for decomposition depending on the problem setting:
1. “First-schedule, then-break” (Kendall, Knust, Ribeiro, & Urrutia, 2010): At first the team pairings for each round are determined, so which teams have a match against each other on each round. This scheduling phase is then followed by calculating the assignment of home-away patterns for the matches resulting from the first step.
This first approach is preferred when there are important requirements for the tournament that have nothing to do with the home-away pattern. Some examples are attractiveness constraints, a number of predefined fixed matches or a reduction in carry-over effects (Trick, 2001).
Trick (2001) solved the first phase by using two different constraint programming approaches. Both the approaches were able to find opponent schedules fairly quickly for single round-robin tournaments up to 20 teams. Adding the requirement to spread the carry-over effects as evenly as possible over the teams made the problem more difficult and the run times increased substantially for 10 teams.
The objective when assigning the home-away patterns in the second stage is often to have the smallest possible number of breaks. In the first phase, Brouwer, Post, & Woeginger (2008) generate an opponenent schedule for an even number of teams n by fixing n/2 matches in every round (n-1 rounds). For the calculated opponent schedule they then try to find the home- away assignment with a minimization of the breaks. For this second phase, Trick M. A. (2001) used an integer programming formulation and was able to solve instances up to 20 teams in relatively low computation times.
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2. “First-break, then schedule” (Kendall et al., 2010): First the home-away pattern for each team is set-up, often with the goal to minimize breaks. Afterwards, the pairings of teams are fixed corresponding to the determined home-away pattern. Placeholders might be used to establish the pairings which are then assigned to the actual teams.
This approach is used for scheduling the Danish triple round-robin tournament by decomposing the problem in four steps. In the first step the home-away patterns are generated to find the best pattern with regards to the number of breaks. In step 2 the teams are assigned to the patterns by solving an integer programming problem. The third step checks if the patterns are feasible. Several feasibility checks are used to find why a pattern set is not feasible. If a feasible pattern set is found, the last step will determine the timetable (Rasmussen, 2008).
2.5.2. Integer Programming Several round-robin tournament schedules have been generated by setting up an integer program model and using a solver to find a solution. As mentioned above, integer programming is often used in decomposition to solve one or more of the subproblems, sometimes combined with other solution methods (constraint programming, heuristics, etc..). Common integer programming methods used in real life or theoretical sport scheduling problems are branch and bound, branch and cut, column generation and Benders decomposition (Kendall et al., 2010).
2.5.2.1. General model Since integer programming models are often used in practice to solve real life scheduling problems with the same characteristics, several authors like Trick, (2003) and Briskorn & Drexl (2009) developed base models that serve as a starting point for more complicated round-robin scheduling problems.
Basic Model The basic elements in this general model for k round-robin tournaments are as follows:
The binary decision variable for a model with n teams T and k(n-1) rounds R is:
ퟏ, 풊풇 풕풆풂풎 풊 풑풍풂풚풔 풂 풉풐풎풆 품풂풎풆 풂품풂풊풏풔풕 풕풆풂풎 풋 풊풏 풓풐풖풏풅 풓 풙 = 풊,풋,풓 ퟎ, 풐풕풉풆풓풘풊풔풆
for teams i,j=1,..,n with i ≠ j and rounds r=1,.., k(n-1).
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The two following basic constraints assure that each team meets every other team exactly k times over the tournament and that each team plays exactly one match per round, respectively:
∑풓∈푹 풙풊,풋,풓 + 풙풋,풊,풓 = 풌 ∀풊, 풋 ∈ 푻, 풊 ≠ 풋
∑풋∈푻\{풊} 풙풊,풋,풓 + 풙풋,풊,풓 = ퟏ ∀ 풊 ∈ 푻, 풓 ∈ 푹
General objective function Since the integer programming models use objective functions to represent the goal they want to achieve, a cost minimization function is used as a general objective function for various problems:
The general cost objective function aims to minimize/maximize the summation of the total cost:
min/max ∑∑풊∈푻 풋∈푻\{풊} ∑풓∈푹 풄풊,풋,풓 풙풊,풋,풓
The cost ci,j,r is an abstract concept but can be used to represent various real-world requirements. For example, the preference for teams to play at home on certain rounds, the economic value of attendance on certain rounds in a maximization approach, etc.
Break constraints The most common goal concerning the home-away patterns is a minimization of the number of breaks. Additional constraints with the following elements are used to represent this in the basic integer programming model:
An additional binary decision variable to represent the breaks:
ퟏ, 풊풇 풕풆풂풎 풊 풉풂풔 풂 풃풓풆풂풌 풊풏 풓풐풖풏풅 풓 풃풓 = 풊,풓 ퟎ, 풐풕풉풆풓풘풊풔풆
for teams i=1,..,n and rounds r = 2,.., k(n-1).
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Two additional constraints that force the decision variable bri,r to be equal to 1 if team i has a home or away-break on round r, respectively. Note that r starts from round 2 (R≥2) as a break in the first round is not possible: