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An Integer Programming Formulation for a Fair Tournament Scheduling in Teamgym in Iceland

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An Integer Programming Formulation for a Fair Tournament Scheduling in Teamgym in Iceland An integer programming formulation for a fair tournament scheduling in TeamGym in Iceland by Andrea Valdimarsdóttir Thesis of 30 ECTS credits submitted to the School of Science and Engineering at Reykjavík University in partial fulfillment of the requirements for the degree of Master of Science (M.Sc.) in Engineering Management June 2021 Supervisor: Eyjólfur Ingi Ásgeirsson, Ph.D., Supervisor Associate Professor, Reykjavík University, Iceland Examiner: Rannveig GuØmundsdóttir, M.Sc., Examiner Head of AGR Services, AGR Dynamics, Iceland i Copyright Andrea Valdimarsdóttir June 2021 ii An integer programming formulation for a fair tournament scheduling in TeamGym in Iceland Andrea Valdimarsdóttir June 2021 Abstract Organizing a sports tournament schedule can be challenging and time-consuming. In this project, the process of generating a TeamGym tournament schedule is automated. Today, the tournaments at FSÍ are made manually. By automating the process, the time for making the tournaments at FSÍ should be considerably shortened and reduce the risk of errors. The main goal of this project is to create a mathematical model that ensures a fair tournament for all teams while fulfilling specific requirements such as all teams must compete once on each event, each round should preferably contain all types of events and care must be taken that teams’ rest periods are within the agreed time frame. The results of the project show that a fair and feasible solution can be found with automation that decreases the scheduling time and ensures a fairer and better-organized tournament. Keywords: Tournament schedule, gymnastics, TeamGym, optimization model. iii Heiltölubestunarlíkan sem b˝r til sanngjarnt mótsplan fyrir hópfimleika á Íslandi Andrea Valdimarsdóttir júní 2021 Útdráttur ùaØ getur veriØ krefjandi og tímafrekt aØ útbúa íΩróttamótsplön. Í Ωessu verkefni er ferliØ viØ aØ útbúa hópfimleikamótsplan gert sjálfvirkt. Í dag eru fimleikamótin hjá FSÍ gerØ handvirkt en meØΩví aØ gera ferliØ sjálfvirkt ætti tíminn viØ gerØ fimleikamóta hjá FSÍ aØ styttast til muna ásamt Ωví aØ minnka líkur á villum. MarkmiØΩessa verkefnis er aØ útbúa stærØfræØilíkan sem tryggir sanngjarnt mót fyrir öll liØ en á sama tíma Ωarf aØ fara eftir ákveØum skilyrØum. Dæmi um slík skilyrØi eru aØ öll liØΩurfa aØ keppa einu sinni á hverju áhaldi, hver umferØ á helst aØ innihalda allar gerØir áhalda og passa Ωarf aØ hvíldartími hvers liØs sé innan ákeØins tímaramma. NiØurstöØur verkefnisins gefa til kynna aØ hægt er aØ búa til sanngjarnari og hagkvæmari lausn sem er sjálfvirk og tekur mun styttri tíma aØ útbúa. Efnisor: Mótsplan, fimleikar, TeamGym, bestunarlíkan. iv Acknowledgements I want to thank my supervisor Dr. Eyjólfur Ingi Ásgeirsson, for the support, meetings, and guidance throughout this project. I would also like to thank Sólveig Jónsdóttir, the CEO of FSÍ, and Íris Svavarsdóttir, former tournament planer at FSÍ, for their cooperation and meetings. Furthermore, I would like to thank my family, friends, and fellow students for the support throughout my time at Reykjavik University. v Contents Acknowledgements v Contents vi List of Figures viii List of Tables ix 1 Introduction 1 2 Background 2 2.1 Artistic gymnastics . 2 2.1.1 Disciplines in artistic gymnastics . 2 2.1.2 Levels in artistic gymnastics . 5 2.1.3 Scoring in artistic gymnastics . 5 2.2 TeamGym . 6 2.2.1 TeamGym events . 6 2.2.2 TeamGym levels . 8 2.2.3 TeamGym scoring . 8 2.3 The TeamGym Scheduling Problem . 9 2.4 The Current Scheduling Process . 9 3 Literature Review 11 3.1 Scheduling problems . 11 3.2 Fairness in scheduling . 12 4 Model 13 4.1 General features of the model . 13 4.1.1 Sets . 13 4.1.2 Parameters . 14 4.2 Variables . 14 4.3 Constraints . 16 4.3.1 Hard constraints . 16 4.3.2 Soft constraints . 16 4.4 Objective function . 18 4.5 The model . 19 4.6 Model adaptability . 20 4.6.1 Changes to the model . 20 5 Data 22 vi 5.1 Data collection . 22 5.2 Data process . 22 5.3 Model input . 23 6 Results 25 6.1 Model results . 25 6.1.1 Results from test case one . 25 6.1.2 Results from test case two . 27 7 Discussion 29 8 Conclusion 32 Bibliography 33 vii List of Figures 2.1 Vault [3] . 3 2.2 High bar [4] . 3 2.3 Even parallel bars [5] . 3 2.4 Uneven parallel bars [3] . 3 2.5 Floor exercises [6] [7] . 4 2.6 Balance beam [9] . 4 2.7 Pommel horse [3] . 4 2.8 Rings [3] . 5 2.9 Balance movement in floor exercises [3] . 7 2.10 Tumbling [16] . 7 2.11 Trampet with vault [3] [17] . 7 2.12 Schedule from the 2nd Cup tournament in 2021 . 10 5.1 Flow of the data process . 23 7.1 Comparison of penalty scores per team for both test cases and the old FSÍ tournament . 31 viii List of Tables 2.1 Women’s and men’s disciplines in artistic gymnastics . 3 2.2 Age range of each level and relevant tournaments [8] [10] . 5 2.3 Maximum score for each level [12] [11] . 6 2.4 Highest score in men’s and women’s free routines from the Cup tourn. in 2020 [13] ........................................ 6 2.5 Women, men and mixed teams compete on same events in TeamGym . 8 2.6 Categories, levels and divisions in TeamGym [18] . 8 2.7 Maximum score for each rating [18] . 9 2.8 Top 3 senior women’s teams from the Cup tournament in 2020 [13] . 9 2.9 TeamGym tournaments [18] . 10 5.1 List of teams competing . 23 5.2 Other data needed for the model . 23 5.3 Weight factors for TeamGym tournaments . 24 6.1 Number of violated constraints for test case one . 26 6.2 Total penalty score for test case one . 26 6.3 Tournament schedule for test case one (due to limited space the table has been divided) . 27 6.4 Number of violated constraints for test case two (same results for both parts) . 28 6.5 Penalty score for part one in case two . 28 6.6 Penalty score for part two in case two . 28 6.7 Tournament schedule for test case two, part one . 28 6.8 Tournament schedule for test case two - part two . 28 7.1 An old FSÍ tournament schedule - part one . 30 7.2 An old FSÍ tournament schedule - part two . 30 7.3 Penalty scores from an old FSÍ tournament - part one . 30 7.4 Penalty scores from an old FSÍ tournament - part two . 30 ix x Chapter 1 Introduction This project will cover the analysis for an integer programming formulation for fair tour- nament scheduling of TeamGym in Iceland. Creating a suitable sports schedule can be a daunting task and, it requires a lot of preparation, logistical planning, and a vital understand- ing of the sport itself. Furthermore, the plan should meet as many requirements and wishes as possible from the Icelandic Gymnastics Association (FSÍ) and other stakeholders. Gymnastics in Iceland is on the rise, and today the sport is the third most practiced in Iceland after football and golf. Furthermore, gymnastics has the second most participants from ages 17 and younger and the most practiced sport for women in general [1]. There are 28 gymnastics clubs operating in Iceland. The biggest clubs are; Gerpla, Stjarnan, Fjölnir, and Grótta. Gymnastics can be practiced individually in artistic gymnastics or as a team, which is called TeamGym. A lot of time has been spent by FSÍ personnel planning tournaments. A few years ago, they tried to improve their methods due to time consumption. They created an organized Excel document that greatly facilitated their work. However, FSÍ employees are still handling the tournament planning by hand. There has been little concern about fairness between teams as FSÍ’s primary goal has been to create as short of a tournament as possible. As the sport has been proliferating in recent years, clubs are sending more and more teams to each tournament, making it more challenging to organize the competitions. Therefore, changes to the tournament structure are necessary, and therefore it is ideal for creating a model that gathers teams and divides them into different rounds for the tournament schedule. The model will save time and reduce costs for FSÍ as the employees can use their time for other things while the model is working. The research objective for this project is to create a mathematical model that generates a fair tournament schedule for FSÍ. To ensure fairness amongst teams in TeamGym tourna- ments, the following needs to be examined; the competition order of events and whether or not the break between rounds is too short or too long. The overall aim is to automate gener- ating tournament schedules by minimizing scheduling time, reducing errors, and meeting as many requirements as possible while delivering a fair, feasible solution to all competitors. Chapter two provides basic information about TeamGym and artistic gymnastics, its events, levels, and scoring. In chapter three, a review of the most relevant literature will be put forth. Chapter four presents the mathematical model and the model adaptability. A brief description will be given of the data, data collection, and data process in chapter five. Chapter six presents the model results and analysis. Chapter seven is the discussion chapter, and in chapter eight, the conclusion can be found. Chapter 2 Background There are many versions of gymnastics, but in Iceland, mainly two are practiced, artistic gymnastics and TeamGym. Artistic gymnastics is practiced individually, but TeamGym is in a team setting. In this chapter, both TeamGym and artistic gymnastics will be explained along with their disciplines/events, difficulty levels, and scoring.
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