Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2, 2366-2374 2025 Publisher: Learning Gate DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 25 December 2024; Revised: 5 February 2025; Accepted: 14 February 2025; Published: 28 February 2025 * Correspondence: gkfkaakstp@gmail.com Alternative approaches to game scheduling of multiple round-robin tournament: Korea professional soccer league (K League 1) in focus Dae-Gon Ko¹, Ha-Ram Kim²*, Tae-Ho Kim³ ¹Bachelor of Science in Biological Science, Yongin University, South Korea; superhero9704@gmail.com (D.G.K.). ²Master of Science in Global Business, Georg-August-Universität Göttingen, Germany; gkfkaakstp@gmail.com (H.R.K.). ³Department of Business Administration, Incheon National University, South Korea; tacokim@gmail.com (T.H.K.). Abstract: This study evaluates the quality of the current schedule of a professional soccer league, K League 1 (Korea Professional Soccer League), which is regarded as the top Asian professional soccer league and has unique characteristics compared to European leagues, such as odd numbers of games between some teams and final rounds based on previous performance. It assesses the total travel distance of teams to play all scheduled games, match fairness, and attendance fairness. Match fairness implies that each team should have the opportunity to play against other teams under similar performance conditions. Teams can be under different performance conditions during a long season due to several reasons, such as injuries and the level of team spirit. Attendance fairness relates to the maximum number of consecutive home or away series for each team. Alternatives that prioritize the minimization of the total travel distance while satisfying other factors are presented. The requirements of soccer league game scheduling differ in certain aspects from those of other sports leagues, like baseball leagues. Moreover, minimization results are compared with the 2023 K League 1 schedule and schedules generated by tournament scheduling methods, including the English Tournament Scheduling approach, where teams face previous opponents in reverse order. Despite extensive research, this study develops a new integer programming model and applies an existing heuristic to K League 1. Using quantitative methods, our research demonstrates match generation and adjustment issues, highlighting significant room for improvement in soccer scheduling. Keywords: Fairness, K league soccer game scheduling, Large-scale integer programming model, Multiple round-robin tournament, Travel distance. 1. Introduction Soccer (Football) is an interesting sport that is considerably popular all over the world unlike baseball and basketball which are played in some specific countries in North America, the Caribbean, Asia, and Europe. The K League, which currently has 12 teams in K League 1 and 13 teams in K League 2, is the first-ranked professional soccer league in Asia of 47 leagues. The league plays once or twice a week from the end of February till the end of October, and teams play a total of 38 round games during the regular season. The first 33 rounds are played on a home-and-away basis (3 games between each pair of teams), and 12 teams are decomposed into group A and group B based on the result of 33 round games and the final 5 round games will be performed within each group. After playing all 38 rounds, the 12th ranked team will be directly go down to K League 2 in next season and 10th and 11th-ranked teams should play the 2nd and the 3rd ranked teams of K League 2 to decide staying in K League 1 or going down to K league [1]. Kim [2] proposed an optimal approach to game scheduling for multiple round-robin tournaments, focusing on the Korea Professional Baseball League (KBO). This study references various findings from 2367 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate Kim’s research to discuss different aspects of schedule optimization. In 2022, the K League had approximately 1.1 million attendees [3] the estimated total sales of all teams were approximately 710 million dollars [4] and the estimated total salary of players was approximately 91 million dollars [5]. In 2011, the economic ripple effect was approximately 590 million dollars [6]. The ability for fans to follow the league's standings and cheer on their favorite teams' games each week is one of the reasons the K League 1 is so popular and has such a significant economic impact in Korea. It is crucial to schedule games so that the players stay in peak physical condition and the audience can continue to pay attention to the league in order to sustain the K League 1's popularity and economic impact. Studies in ice hockey, football, volleyball, and soccer have concentrated on the number of breaks that show how frequently all teams play two home or two away games in two consecutive series [7-9]. because travel distance does not change in any kind of scheduling schemes. In fact, direct travel from an away venue to the other away rarely happens in these sports. However, research on baseball and basketball game scheduling has centered on the objective function of each team's travel distance [10- 13]. When compared to other sports leagues, baseball and basketball leagues feature a significantly higher number of series or games and teams should travel frequently from one away venue to the other away venue directly without coming back home venue. Match fairness is another crucial component. This suggests that every team ought to be given the chance to play other teams in comparable performance scenarios. Throughout a lengthy season, teams may experience varying performance circumstances due to a variety of factors, including injuries and team spirit levels. Therefore, depending on when each team is scheduled to play other teams, match fairness may be broken. As a result, this element influences the teams' final standing at the end of the season. In fact, during a K League 1 season, a squad may win seven straight games then lose seven more. Each team must play exactly one game against every other team before playing against the same team again, and there should be a fair amount of time between matches to improve match fairness. In addition to these, the cap on the number of consecutive home or away series for each team should be taken into account in order to reflect attendance equity. This research offers valuable information regarding the K League 1 game schedule. By enhancing current models, which may be large-scale optimization issues into certain small-scale sub-problems to reach solutions to an NP-hard problem, we create an integer programming model that captures the distinctive features of the K League [14] and a commercial solver is used to solve the model directly; in some situations, a problem-specific heuristic may be taken into consideration. To propose a better game schedule to the K League 1, the results are compared in terms of the total travel distance, match fairness, and attendance fairness. Moreover, we will propose to the K League 1 the use of the game scheduling scheme from this study for future seasons. Table 1 summarizes the literature on round-robin tournament and the location of this research. 2368 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate Table 1. Summary of literature on round-robin tournament [15]. In Section 2, we go over the specifics of multiple round-robin competitions, especially those held by the KBO League, and present an integer programming model that takes into account all of the previously mentioned variables. In Section 3, the integer programming model is used to try to solve the problem directly. If the model is unable to solve the problem directly, a problem-specific heuristic is taken into consideration. Section 4 presents and explains the empirical findings and some of the issues they raise. The K League 1's current schedule is derived from the English scheduling approach in Section 5, and the schedules produced by our model and heuristic are then contrasted to offer fresh managerial perspectives. The paper is finally summarized in Section 6. 2. Notation and Model Description Reducing the overall travel distance for every team is the main challenge in this study. Based on Kim's basic model, we create a linear integer programming model for a multiple round-robin tournament problem [15]. To accomplish the foregoing, a specific number of teams, n (𝑖, 𝑗 = 1, . . . , 𝑛) and the number of games between each pair of teams k are available. We have to define rounds (𝑟 = 1, . . . , 𝑅), where R is equal to 𝑘 × (𝑛 − 1). In the K League 1, there are 12 teams (n = 12), with each team playing three matches against every other team (k = 3), across a total of 33 rounds (R = 33). The parameters H₁ and A₁ represent the maximum number of consecutive home and away series, respectively, that a team can have. The maximum number of home series within any 11-round period is denoted as H , which is calculated as: �̄� = [ 11 2 ] = 6 Approach Application Direct IP Two-phases based on IP Simulated annealing/Tabu search Other heuristics Theory Introductory /Survey Baseball Kim and Kim [15] Russell and Leung [10] and Easton, et al. [11] Anagnostopoulos, et al. [16] Cain [17]; Hoshino and Kawarabayashi [13] and Kim and Kim [15] De Werra [7]; Trick [18]; Briskorn and Drexl [19]; Nurmi, et al. [20] and Thielen and Westphal [14] Anderson [21]; Easton, et al. [11]; Easton, et al. [22]; Rasmussen and Trick [23] and Kendall, et al. [24] Football/ Soccer Durán, et al. [25]; Fiallos, et al. [26] and Ribeiro [27] This research Ribeiro and Urrutia [28] Rasmussen and Trick [23] and Rasmussen [29] This research Ice Hockey Kyngas and Nurmi [9] Nurmi, et al. [20] and Nurmi, et al. [30] Basketball Bao [12] and Durán, et al. [31] Wright [32] Volleyball Bonomo, et al. [33] 2369 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate This constraint ensures a balanced distribution of home and away matches throughout the season. Regarding travel distances, the variable ijd represents the distance in kilometers between the home venues of teams i and j . The total travel distance of all teams is used as the objective function, aiming to minimize overall travel. To model this scheduling problem, we define decision variables as follows. The binary variable , ,r i jx is assigned a value of 1 if team i hosts a home match against team j in round r , and 0 otherwise. The linear integer programming model of the K League 1 is as follows: ( ) ( ) , , , , , , 1 1 1r i j R n n r i j r j i ij ji x r i j Min x x d d = = = +  + (1) subject to, , , 0, 1,..., , 1,...,r i ix r R i n= = = (2) , , , , 1 1 1, 1,..., , 1,..., n n r i j r j i j j x x r R i n = = + = = =  (3) 1 2 2 , , , , 1 1, 1,..., , 1,..., n n r i j r j i r r n x x i n j n − − = = + = = =  (4) 2 2 3 3 , , , , 2 1 1, 1,..., , 1,..., n n r i j r j i r n r n x x i n j n − − = = − + = = =  (5) …… ( 1) ( 1) , , , , ( 1) ( 2) ( 1) 1, 1,..., , 1,..., k n kkn k r i j r j i r k n k r kn k x x i n j n + − +− = − − − = − − + = = =  (6) 1 , , 1 , 1,..., , 1,..., n r i j r x H i n j n − =  = = (7) 2 2 , , , 1,..., , 1,..., n r i j r n x H i n j n − =  = = (8) …… , , ( 1) ( 2) , 1,..., , 1,..., kn k r i j r k n k x H i n j n − = − − −  = = (9) 1 1 2 , , 1 , 1,..., 2, 1,..., r n r i j C r r j x H r R i n + = =  = − = (10) 1 2 2 , , , , 1 1, 1,..., , 1,..., n n r i j r i j r r n x x i n j n − − = = + = = =  (11) 2 2 3 3 , , , , 2 1 1, 1,..., , 1,..., n n r i j r i j r n r n x x i n j n − − = = − + = = =  (12) …… 2370 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate ( 1) ( 1) , , , , ( 1) ( 2) ( 1) 1, 1,..., , 1,..., k n kkn k r i j r i j r k n k r kn k x x i n j n + − +− = − − − = − − + = = =  (13) ( ) 1 1 1 3 , , , , 1, 1,..., 3, 1,..., , 1,..., r r i j r j i r r x x r R i n j n + = +  = − = = (14) As seen by the objective function (1), the goal is to reduce the overall journey distance. The choice variables are specified as binary variables by constraint (2). In a round, every team faces exactly one other team, according to constraint (3). According to constraints (4), (5), and (6), every team competes against every other n-1 team in every n-1 round. Constraints (7), (8), and (9) restrict how many home games are allowed in each n-1 round. (10) restricts how many home games may be played in a row. Each pair of teams alternates between home and away games, as stated in constraints (11), (12), and (13) respectively. According to constraints (14) every pair of teams must play two games in a row with a minimum of four rounds between them. 3. Empirical Data The K League 1 schedule for 2023 provides the data for our empirical investigation. For forty years, the league has been in existence. Depending on the outcome of the 33 rounds, each team plays both 33-round and 5-round games. The locations of the 12 Korean teams are displayed in Figure 1. Google Maps was used to calculate the distances (in kilometers) between each pair of home venues for every team. The data is summarized in Table 2. The greatest distance between two home venues is roughly 661 kilometers, as seen in the table. 3.5 kilometers is the minimum distance. The standard deviation of 151.02 km is high, and the mean distance of 266.61 km is roughly half of the greatest distance. It makes sense given each person's overall trip distance. It makes sense that each team's overall travel distance would vary, and the K League 1 would try to cut down on both this distance and the variation across teams. Table 2. Summary of distances between each pair of home venues (in kilometers). Min Max Mean Standard Deviation 3.5 661 266.61 151.02 Figure 1. Venue of 12 teams in K League 1. 2371 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate 4. Results and Discussion As noted in previous Sections, we develop two additional game schedules to the 2023 real schedule: one is generated from the integer programming model in section 2 and the other is generated from the English tournament scheduling approach, and compare them with each other in the aspects of total travel distance, match fairness, and the length of consecutive home and away games of teams. Table 3 shows us the total travel distance of teams of the three schedules. As seen, the total travel distance and travel distance of each team are equal in all 3 schedules though they give teams different game schedules. This result is very natural because all teams visit the away venue and come back home before the next game without going to the next away venue directly unlike baseball league. In fact, soccer league does not play every day while baseball league does. Therefore, any schedule will not make a difference in travel distance. The difference in travel distance across teams comes from the location of home venues of teams as illustrated in Figure 1. Table 3. Total travel distance of K league 12 teams (km). Team 2023 schedule IP schedule English schedule 1 18,378 18,378 18,378 2 17,748 17,748 17,748 3 15,966 15,966 15,966 4 13,236 13,236 13,236 5 15,126 15,126 15,126 6 13,581 13,581 13,581 7 13,623 13,623 13,623 8 20,586 20,586 20,586 9 15,744 15,744 15,744 10 15,078 15,078 15,078 11 33,156 33,156 33,156 12 18,936 18,936 18,936 Total 211,158 211,158 211,158 Mean 17,597 17,597 17,597 Standard deviation 5418.21 5418.21 5418.21 Table 4 and Figure 2 illustrate the result of match fairness across teams measured by the match interval of each pair of teams. Because the English schedule has a minimum of 1 and a maximum of 21 as match intervals, it is not good in the aspect of match fairness though the average is equal to other schedules. As seen in the top panel of Table 4 and Figure 2, because the summary of the average match interval of each pair of teams is homogeneous though the average match interval of each pair of teams is heterogeneous across 3 schedules, there is no difference in match fairness found. However, if we look at the second and third panels of Table 4 it is possible to find a slight difference in match fairness between 2023 real schedule and the optimal schedule from our model. The second panel illustrates the average match interval with the strongest 4 teams in the previous season and the optimal schedule seems to be slightly better in that the average match interval with strong team is longer than and standard deviation is shorter than the 2023 real schedule though it is not statistically significant. The third panel shows us the nonparametric runs test result. The optimal schedule has slightly more runs than 2023 real schedule, which means that long match interval and short match interval repeated more frequently, and the optimal schedule has more observations below the mean than the latter, which means that the optimal schedule has more observations around mean than 2023 real schedule. In conclusion, the optimal schedule is said to be better than the 2023 real schedule in the aspect of match fairness. The p-values show that average match intervals are random in all three schedules because they are greater than 0.05. 2372 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate Table 4. Average match interval of each pair of teams. Average match interval Current schedule Optimal schedule English schedule Mean 9.33 9.33 9.33 Standard deviation 1.06 1.06 1.06 Min 7.67 7.67 7.67 Max 11.00 11.00 11.00 Average match interval with top 4 teams in 2022 season Mean 9.32 9.37 9.32 Standard deviation 1.07 1.06 1.10 Min 7.67 7.67 7.67 Max 11.00 11.00 11.00 Runs test # (expected #) of runs 34 (33.72) 36 (33.72) 41 (33.72) # of obs. above mean vs. below mean 36 vs. 30 30 vs. 36 30 vs. 36 p-value 0.946 0.570 0.069 Figure 2. Average match interval of each pair of teams across schedules. In the number of consecutive home/away rounds, the optimal schedule guarantees that they are less than or equal to 2 while the English schedule has 1/1 and 2023 real schedule has 2/3 of 3 consecutive home/away rounds. In the rule of home-and-away-in-turns, while the optimal schedule and the English schedule keep the rule with no exception, the 2023 real schedule violates the rule 36 times in a season. 5. Conclusions The overall trip distance, match fairness, and attendance fairness were the main topics of this study on soccer game scheduling. To optimize the overall journey distance while accounting for all other 2373 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 2: 2366-2374, 2025 DOI: 10.55214/25768484.v9i2.5089 © 2025 by the authors; licensee Learning Gate variables, an integer programming model was also created. The model's output schedule quality was then contrasted with the English tournament schedule and the K League 1 calendar for 2023. It was verified that the K League 1 schedule produced by our model was significantly better in terms of overall travel distance, match fairness, and attendance fairness. Specifically, our suggested model was also validated to have the ability to provide higher-quality answers for problem settings in less than two hours of CPU time utilizing solvers such as CPLEX. Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. This study followed all ethical practices during writing. Acknowledgment: The authors sincerely acknowledge the financial support provided by the Yongin University Alumni Association Scholarship Foundation for this research. This support, granted through the 2023 Research Funding Program, enabled participation in the IFORS 2023 conference and contributed to the development of this study. Copyright: © 2025 by the authors. This open-access article is distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). References [1] K. League, "K League official website," Retrieved: http://www.kleague.com, 2021. [2] T. 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