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Dept. of Mechanical & Industrial Engineering Concordia University, Montreal, Canada Professor Dr. Hossein Hashemi Doulabi

Using Operation Research Techniques to Schedule Germany’s Professional Soccer League November 28th, 2018

Team members: Hozaifa Dekis Karan Hirpara Livia Menezes Mohamad Sadek Vivek Mendapara

TABLE OF CONTENTS Abstract ......................................................................................................................................................... 2 1. Introduction ............................................................................................................................................... 3 1.1 Problem Definition & Assumptions.................................................................................................... 3 2. Literature Review...................................................................................................................................... 5 3. Modeling ................................................................................................................................................... 6 3.1 Objective Function .............................................................................................................................. 7 3.2 Constraints .......................................................................................................................................... 8 3.2.1 Schedule Constraints............................................................................................................ 7 3.2.2 Sequence Constraints ........................................................................................................... 7 3.3 Decision Variables & Notation ........................................................................................................... 8 3.4 Objective Function Formulation ......................................................................................................... 8 3.5 Constraint Formulation ....................................................................................................................... 9 4. Solution Approach .................................................................................................................................. 10 4.1 Interpreting CPLEX Output .............................................................................................................. 10 5. Computational Results And Analysis ..................................................................................................... 10 5.1 Analyzing Different Options............................................................................................................. 10 5.2 First Feasible Solution Case.............................................................................................................. 11 5.3 Second Feasible Solution Case ......................................................................................................... 14 6. Conclusion .............................................................................................................................................. 16 Bibliography ............................................................................................................................................... 17 Appendix ..................................................................................................................................................... 18

LIST OF TABLES 1. Table 1: 2018 Bundesliga Team Listing ............................................................................................ 12 2. Table 2: First Proposed Solution Schedule ....................................................................................... 13 3. Table 3: Second Proposed Solution Schedule ................................................................................... 15

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Abstract Germany’s professional soccer league Bundesliga league is an annual sport event tuned in and enjoyed by millions across the globe. Besides, this national league is known particularly for having the highest average stadium attendance worldwide. Similarly, to other sports event, developing its matches scheduling considering the sequence and order of the matches between the teams is an essential component for both teams of each match, their sponsors and fans. This project aims to reach a feasible and superior master matches schedule using Operational Research analytical techniques and development modeling tools. The mathematical models will be developed by using CPLEX software and its outputs will be analyzed as potential solutions.

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1. INTRODUCTION Football, also known as soccer, is a sport that spreads union among nations beyond boundaries, especially when it comes to international football leagues of the world. The teams involved in national leagues or international championships spend a high amount of fortune with football starts, as well as the clubs bringing all the best skills into one team builds the perfect recipe for the best show of football. Setting the entertainment and sports’ emotional aspect aside, the whole sports business demands a management need for a tool to schedule and organize these leagues and, thus, a scientific technique to build a reliable management structure, since football has stopped being a mere entertainment source to become a business, which involves profit for sponsors and partners companies, players and people who invest money on their teams. For such management, there must be an optimal schedule for each regular season of the league, considering the game calendar that must satisfy the fairness of it to the participating teams, that also must be economically beneficial for every involved part, as both teams and fans. Such scheduling would be nearly impossible if done by hand, which casts light for the need of an efficient scheduling method for such great event. 1.1. Problem Definition & Assumptions The scheduling of a professional German soccer league using operation research techniques will be performed on Bundesliga championship, which is the most tuned in sport event in the world of soccer. There are eighteen different teams participating of this yearly basis league, and they play against each other along the year. The season starts in August and it runs until May. Each team

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play 34 matches which includes home and away matches against the other seventeen teams. Thus, the whole championship consists of a total of 306 matches. Along the league, there’s a point system that is based on the results of the previous matches, and it determines the ranking of the current participating teams. If a match results in a win for a given team, this team earns 3 points as an outcome of it. If it’s a draw, the outcome will be 1 point, and finally a defeat will return no points to the loser team. This league is structured such a way that the final results will be determined according to the obtained points of each teams with no Quarters, Semi or Final matches. Each team will face one Home match and one Away match with the remaining 17 teams, which means each team will play 34 matches, 17 matches should be held on its own ground and the remaining 17 matches should be held on the other team’s ground, and the one with the higher amount of points will be named the champions of the year. The tie-breaker criterion if there are two teams with the same amount of points the difference of goals number of each team. Finally, as operations research (OR) involvement in scheduling sports events has grown significantly due its increase of popularity of such events. Given that the German Bundesliga League’s structure and constraints offers the perfect opportunity for the application of OR, the project will focus on creating a master matches schedule for the German Bundesliga League by solving a set of objective function and constraints using CPLEX software. The project will take into consideration some of the issues encountered by previous years’ schedules, for instance, by trying to decrease the travelling time of one team as compared to its opponents for a set of consecutive games during previous seasons. Additionally, some constraints were added to the model the scheduling process such as the concentration of the matches between two teams from

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certain regions they belong, among other constraints that should be explained along the project report. 2. LITERATURE REVIEW As the sport business is getting significant boom every year, effective scheduling of sports events is happening on regular basis all over the world, for example the scheduling of the football and basketball leagues of the United States of America is happening on a regular basis every season. This scheduling is either done by the team management or by a specialized third party with the objective of designing an efficient season schedules, currently there are some software’s and applications that is developed specifically to schedule sports events. An example of that application as mentioned by the authors of “Reducing travel costs and player fatigue in the national basketball association” a total savings of $757,000 or 20% of costs over the NBA’s current schedule has been achieved by developing a new model that optimize the travelling cost of the participating teams during the seasons (Bean). There are other reports pertaining scheduling of sports events published in 1997 with titles “Scheduling a major college basketball conference” and “Scheduling a major college basketball conference-revisited” presented a valid method to scheduling with programming (Nemhauser). The authors of the reports used a finite-domain constraint programming technique to complete the scheduling (Henz). In the past few years, the existence of sporting scheduling literature related academic publications had significantly increased. The new articles have been published in this field highlighting some new aspects of un famous leagues that can be a potential of future research topic.[4, 5, 6]

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Interests in the field of sports scheduling using Operation Research tools peaked when an article titled “Travelling Tournament Problem” was published. The developed algorithms focused on the minimizing the distances teams in the league should travel. Although they have not developed the algorithm by considering the real case setup, the authors have developed range of methods and algorithms. In the article “Solving the travelling tournament problem: a combined integer programming and constraint programming approach” a combination of integer programming and constraint is used as a method of finding the optimal solution for leagues of up to 8 teams (Easton). In the paper “Heuristics for the mirrored traveling tournament problem”, heuristics are developed for the mirrored type of the travelling tournament problem for tournaments that has same order of games in each rounds (Rebeiro). The authors for “A simulated annealing approach to the traveling tournament problem” proposed solving

the

travelling

tournament

problem

based

on

simulated

annealing

(Anagnostopoulos). Whereas in a taboo search, application is utilized for the same purpose. Another interesting publication is related to the matches’ concentration in specified location with title “Scheduling the Chilean soccer league by integer programming” in 2012 (Cardemil) (Dur´an). 3. MODELING 3.1 Objective Function The objective function’s goal is to minimize the occurrences of a Northern team travelling to a Southern team’s field, or vice versa, during the rounds 11 to 17. The objective function is minimizing the variable ฀฀฀฀฀฀฀฀ while the model also decides on the values of both ฀฀฀฀฀฀ & ฀฀฀฀฀฀ , as will be shown in the next sections. All these three variables are binary values, so the final solution will be a corresponding table of each period and the related decision variables. 6|Page

3.2 Constraints The Bundesliga, has its own set of governing rules, but many of the league’s and regulations are set by UEFA (Union of European Football Associations) and overseen by FIFA (Fédération Internationale de Football Association) as well. The following constraints will mimic such regulations imposed on the league. 3.2.1 Schedule Constraints These constraints are as follows: o Each team should play against the other 17 teams o Each team has to play the match at its home or at the opponent’s home ground. o Each team must play at least 8 of the 17 rounds at home while playing the remaining games away. Or, the team must play a maximum of 9 matches of the 17 rounds at home and the rest away. 3.2.2 Sequence Constraint o A team cannot play more than a set of two consecutive games at Home. o A team cannot play more than two consecutive games at Away. o A team from the North group cannot play 2 consecutive Away games against the South group, only one game can be played at South and the other Away game has to be played against a team from the North group and vice-versa.

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3.3 Decision Variables & Notation The following are decision variable and notations: o I set of 18 teams, i & j belongs to I o K set of 17 rounds, i & j belongs to K o N set of the 9 teams of the North group, N ⊂ I o S set of the 9 teams of the South group, S ⊂ I o North group teams are ranked from 1 to 9,and South group teams are ranked from 10 to 18 for all constraints i cannot equal to j. o Team, i, playing against team, j, in round, k, at home. 1; ฀฀฀฀฀฀฀฀ i" is playing with team "฀฀ at round "฀฀" ฀฀(฀฀฀฀฀฀) = � 0; ฀฀฀฀฀฀฀฀

o Team, i is playing two consecutive home games at round, k and k+1. ฀฀(฀฀฀฀) = �

1; ฀฀฀฀฀฀฀฀ "฀฀" ฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀ ℎ฀฀฀฀฀฀ (฀฀, ฀ ฀ + 1) 0; ฀฀฀฀฀฀฀฀

o Team, i is playing two consecutive away games at round, k and k+1. ฀฀(฀฀฀฀) = �

1; ฀฀฀฀฀฀฀฀ "฀฀" ฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀ (฀฀, ฀ ฀ + 1) 0; ฀฀฀฀฀฀฀฀

3.4 Objective Function Formulation The objective function is minimizing the variable ฀฀฀฀฀฀฀฀ , simply put, the number of matches between team i against j from rounds 11 to 17, where i belongs to the northern group set (teams 1 to 9) while team j belongs to the southern group (teams 10 to 18). ฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀: � �

�

฀฀(฀฀฀฀฀฀)

฀฀∈฀฀ ฀฀∈฀฀11≤฀฀≤17

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3.5 Constraint Formulation The following will describe the constraint formulations: o Each team is going to play against other 17 teams only once over the 17 rounds duration: �[฀฀(฀฀฀฀฀฀) + ฀฀(฀฀฀฀฀฀)] = 1; ∀฀฀ & ฀฀ ∈ ฀฀

(1)

฀฀

o Each team is playing the round either home or away: �[฀฀(฀฀฀฀฀฀) + ฀฀(฀฀฀฀฀฀)] = 1; ∀฀฀ ∈ ฀฀ & ∀฀฀ ∈ ฀฀

(2)

฀฀

o Each team cannot play less than 8 games at home and each team cannot play more than 9 games at home: 8 ≤ � � ฀฀(฀฀฀฀฀฀) ≤ 9; ∀฀฀ ∈ ฀฀ ฀฀

(3)

฀฀

o Each team plays at most one sequence of two consecutive games at home: � ฀฀(฀฀฀฀) ≤ 1; ∀฀฀ ∈ ฀฀

(4)

฀฀...