Journal of Avance Research in Social Sciences an Humanities Volume 2, Issue 1 (17-26) DOI: https://x.oi.org/10.26500/jarssh-02-2017-0103 A tournament scheuling moel for National Basketball Association (NBA) regular season games HUA-AN LU, WEI-CHENG CHEN Department of Shipping an Transportation Management, National Taiwan Ocean University, Keelung, Taiwan Abstract Professional sports have become a business market with high rewars in global economies. In particular the high level leagues can attract a group of sport populations over global countries, such as the National Basketball Association (NBA). Tournaments are a main part for the whole operation of the professional sport leagues that can maintain an increase the attractiveness to fans. The NBA currently consists of 30 teams home at ifferent cities in USA an Canaa. The whole tournament of its regular season every year lasts to play nearly 24 weeks. Nevertheless, the tournament scheuling problem is complicate an large-scale an subect to many factors, such as ates, venues, opponents of games, etc. This stuy aims to formulate a mathematical programming moel for the tournament scheuling of the NBA base on the fairness of opponent arrangement. An Integer Programming (IP) moel was propose to minimize the ifference of opponent competitiveness among all teams. Opponent competitiveness is efine as the sums of opponent winning percentages in the last season for every game. Constraints inclue some known regulations an previous practices in publishe ournalism reports. Numerical experiments reveale this moel can obtain promising results for the tournament arrangement of the NBA. Keywors: Tournament Scheuling, National Basketball Association (NBA), Integer Programming (IP), Opponent Competitiveness Receive: 15 August 2016 / Accepte: 22 November 2016 / Publishe: 22 February 2017 INTRODUCTION Nowaays, sports are not ust general leisure activities but have also become a business market with high rewars because of the evelopment of various professional athleticisms. In particular a variety of professional sport leagues attract people over the worl to be their fans, such as the National Football League (NFL), the Maor League Baseball (MLB), an the NBA in the USA. Tournaments of every kin of professional athleticisms can create huge benefits in terms of high ratings of TV relays an large sales of tickets an peripheral proucts. Tournament scheuling is one of the important promotional tools for professional leagues. Fans always notice the tournament arrangement of teams of their star athletes. Tournament fierceness an fairness are main iscusse topics of team management, athletes an fans because the scheules affect the whole season performance for all teams an all athletes. However, tournament scheuling is not an easy task by manual arrangement for a large league like the NBA with 30 teams, as the scheules have to follow numerous an complicate restrictions. It is not to mention to meet the obectives of fierceness an fairness in such rules. We are motivate to propose a tournament scheuling moel with a fair obective for the NBA regular season games. An appropriate scheule arrangement can increase the attractiveness of plays to the fans an even absorb more extensive spectator groups. Furthermore, such arrangement may promote the performance of athletes an reuce their negative impacts on psychological an physiological sies. Teams in the NBA have their own home courts locate at ifferent cities in the USA an Canaa. They are ivie into various conferences an ivisions. Each team plays same an fixe number of total games with even home an away uring certain ays in a regular season. Games competing with teams across conferences an ivisions have a flexible conition. The ecision variables are the venues for home/away teams in certain available Corresponing author: Hua-An Lu Email: vc364114@gmail.com Copyright c 2017. Journal of Avance Research in Social Sciences an Humanities
Lu et al. / A tournment scheuling moel for ates with commercial consierations or without conflict activities in the venues. The purpose of this stuy is to formulate a mathematical programming moel for tournament scheuling of the NBA regular season games. Subect to the scheuling rules, an IP moel was propose to minimize the ifference of opponent competitiveness among all teams. Through some case tests, this stuy attempts to ensure that the robustness of solution results can be applie to the practical programming. The remaining contents of this paper make a literature review regaring sport tournament scheuling in Section 2. Section 3 introuces the tournament scheule rules in the NBA an the propose moel. Section 4 iscusses the results of some test cases. The final section summarizes our stuy an provies some suggestions for the future research. LITERATURE REVIEW Scheuling problems in sports are require such as the arrangement for the tournaments an umpires. Wright (2009) reviewe the works using Operations Research (OR) techniques for about 50 years. Ribeiro (2012) analyze the funamental problems for sport scheuling an provie a survey of applications of optimization methos to scheuling problems in professional leagues of ifferent sport isciplines. The main problems arising in tournament scheuling inclue issues such as breaks minimization, istance minimization, the Traveling Tournament Problem (TTP), an carry-over effects minimization. These problems have been solve by ifferent exact an approximate approaches, incluing integer programming, constraint programming, metaheuristics, an hybri methos. Some efforts are evote to the specific problems. Costa (1995) combine the mechanisms of genetic algorithms an tabu search to solve National Hockey League (NHL) optimization problems, hoping to reuce costs. This hybri metho evelope is well suite for Open Shop Scheuling Problems. Dilkina an Havens (2004) use Constraint Satisfaction Problem (CSP) to scheule 256 games in 17 weeks for 32 teams of the NFL. These teams were ivie into the National Football Conference (NFC) an the American Football Conference (AFC). Some home/away spacing constraints an other relate regulations require to be ahere. An available scheule ha to be use with the US television network live. They provie an overview of the constraint solving methoology employe an the implementation of the NFL prototype system. Duran, Guaaro, an Wolf-Yalin (2012) propose an integer linear programming (ILP) moel to scheule the Secon Division of the Chilean professional soccer league. Geographical restrictions are particularly important because of the special geographical environment of Chile for long bus travel istances. Chilean league officials have successfully use this moel to scheule all five Secon Division tournaments between 2007 an 2010. There were stuies focuse on the nonprofessional basketball games. Nemhauser an Trick (1998) evelope a combination of integer programming an enumerative techniques to scheule nine universities in the Atlantic Coast Conference (ACC). A basketball competition assigne schools to play home an roa games against each other over a nine-week perio. This work was applie to make reasonable scheules in 1997-1998 seasons. Wright (2006) use a variant of simulate Annealing (SA) to scheule games for National Basketball League of New Zealan. This approach was applie to make the scheule in 2004 season. For the NBA, recent research evote to various perspectives, such as statistical analysis for winning playoffs (Summers 2013), ientifying basketball performance inicators in regular season an playoff games (Garcia et al. 2013), time trens for inuries an illness, an their relation to performance (Polog et al. 2015). The stuies for tournament scheuling are relatively limite. Bean an Birge (1980) constructe scheules of NBA base on Traveling Salesman Problem (TSP). The obective is to reuce the total number of passenger miles for the league travels, 22 teams in the NBA at the time. This approach successfully evelope scheules for the 1978-1979 an 1979-1980 seasons with savings of 20.4% costs. Actually, current structure in the NBA has been change than 20 years ago. More teams an more fans make the tournament scheuling problem become more complicate an commercialize. METHODOLOGY This section briefly introuces current teams in the NBA an scheuling rules for their tournaments. A propose mathematical moel can then be formulate with the obective of minimizing the ifference of opponent competitiveness among all teams. Opponent competitiveness is efine as the sums of opponent winning percentages in the last season for every game. 18
Lu et al. / A tournment scheuling moel for Problem Description The NBA has been evelope for nearly 70 years. Initially, there are only 11 teams in NBA, each team playing 60 games in a season. Currently, the NBA has become a huge sport league, with 30 teams home at ifferent cities in USA an Canaa. Figure 1 shows the location of home grouns for these teams. They are ivie into 2 conferences, eastern conference an western conference, an belong to 6 ivisions, respectively. Eastern Conference a. Atlantic ivision inclues Boston Celtics, Brooklyn Nets, New York Knicks, Philaelphia 76ers, an Toronto Raptors. b. Central ivision inclues Chicago Bulls, Clevelan Cavaliers, Detroit Pistons, Iniana Pacers, an Milwaukee Bucks. c. Southeast ivision inclues Atlanta Hawks, Charlotte Hornets, Miami Heat, Orlano Magic an Washington Wizars. Western Conference a. Southwest ivision inclues Dallas Mavericks, Houston Rockets, Memphis Grizzlies, New Orleans Pelicans, an San Antonio Spurs. b. Northwest ivision inclues Denver Nuggets, Minnesota Timberwolves, Oklahoma City Thuner, Portlan Trail Blazers, an Utah Jazz. c. Pacific ivision inclues Golen State Warriors, Los Angeles Clippers, Los Angeles Lakers, Phoenix Suns, an Sacramento Kings. Figure 1. Location of home grouns for all NBA teams Relate scheuling regulations in scheuling tournaments in the NBA are collecte as follows: Each team must play 82 games in a season, 41 home games an 41 roa games. Each team only has 1 game at most in a ay. Each team only has 2 games at most within 3 ays. More teams have to play on Friay an Saturay. Lakers an Clippers use the same court, i.e., Staples Center, so they cannot use the home team ientity to play game on same ay. 19
Lu et al. / A tournment scheuling moel for No games can be playe at the Thanksgiving Day, Christmas Eve, All-Star Game an NCAA finals. A team has to play roa games when its home court cannot be use in a particular perio of time. Each team has to play 4 games (2 home games an 2 roa games) with the other 4 teams in the same ivision. Each team has to play 2 games (1 home game an 1 roa game) with the other 15 teams in ifferent conferences. Each team has to play 3 or 4 games (1 home game an 2 roa game, 2 home game an 1 roa game, or 2 home game an 2 roa game) with the other 10 teams in the same conference but ifferent ivisions. Some special games are arrange on special ates, such as a eliberate game for Warriors an Cavaliers (last season s finals) at Christmas in 2015-16 season. Mathematical Moel The propose moel aims to minimize the ifference of opponent competitiveness among all teams subect to the scheuling regulations mentione in the methoology Section. Before introucing the contents of the moel, notation for this moel is efine as follows: Inices: : Inices for scheuling ays i, : Inices for team Sets: W : the set of ays for Friay an Saturay. H : the set of ays that cannot arrange games. R i : the set of ays that have to arrange roa games for team i. S: the set of ays that have to arrange particular games. E : the teams must play on ay in. X : the set of teams for share home. Decision variables: x i : if team i plays home game with team at ay groun; 1 for yes, 0 otherwise. Z i : opponent competitiveness for team i, i.e., the sum of opponent winning percentages in the last season for every game for team i. Z h : the highest opponent competitiveness among all teams. Z l : the lowest opponent competitiveness among all teams. Parameters: C i : the conference of team i. V i : the ivision of team i. W : the last seasonal winning percentage of team. D: the total ays of a regular season. K: the total number of teams. a: the total number of games for each team playing in a regular season. f: the total number of home (away) games with teams of the same conference an the same ivision. g: the total number of home (away) games with teams of the ifferent conference. h: upper boun of total number of home (away) games with teams of the same conference but ifferent ivisions. l: lower boun of total number of home (away) games with teams of the same conference but ifferent ivisions. m: maximum ays for consecutive plays. n: upper boun of the consecutive games in maximum ays for consecutive plays. q: lower boun of total number of home an roa games with teams of the same conference but ifferent ivisions. r: a ratio of teams playing on Friay an Saturay. As the Equation (1), the obective function of this moel is to minimize the ifference of opponent competitiveness. It can be calculate by the ifference of Z h an Z l. Minimize = Z h Z l (1) 20
Lu et al. / A tournment scheuling moel for Some constraints require to be followe. Opponent competitiveness efinition constraints inclue Equation (2) to (4). Equation (2) is the sum of opponent winning percentages in the last season for every game for team i. Z i = w x i + w x i i (2) Z h Z i i (3) Z l Z i i (4) Equation (5) to (7) are the limitations for games. Equation (5) means that the total number of home games must be equal to roa games. Equation (6) enforces that the total number of games of each team must be the same. Equation (7) limits each team to play only one game at most in a ay. x i = x i i (5) x i + x i = a i (6) x i + x i 1 i, (7) Equation (8) can limit each team to play only n games at most in consecutive m ays. +(m 1) = x i + +(m 1) = x i n i, = 1, 2,..., D (m + 1) (8) If two or more teams use same court, they cannot use the home team ientity to play game on the same ay as Equation (9). x i + x i 1, (i, i )εx (9) Equation (10) enforces that specific ays can play no games. On the other han, certain Friays an Saturays must play games for commercial consierations as Equation (11). Equation (12) can scheule special games on special ates for certain terms. Equation (13) limits a team to play roa games when its home court cannot be use in a particular perio of time. x i + i i x i rk 2 i x i = 0 ε H (10) ε W H (11) x i + x i = 1 ε S, (i, )εe (12) x i = 0 i, ε R i (13) Equations (14) to (22) are constraints playing with opponents in conferences an ivisions. Equations (14) an (15) consier the total number of home (away) games for same conference same ivision. Equations (16) to (17) limit the total number of home (away) games for ifferent conferences. Equations (18) to (22) limit the total number of home (away) games for same conference but ifferent ivisions. Equations (23) an (24) are nature of the variables. x i = f i, i(v i = V ) (14) 21
Lu et al. / A tournment scheuling moel for x i = f i, i(v i = V ) (15) x i = g i, i(c i = C ) (16) x i = g i, i(c i = C ) (17) x i + x i q i, i(c i = C, V i V ) (18) x i h i, i(c i = C, V i V ) (19) x i l i, i(c i = C, V i V ) (20) x i h i, i(c i = C, V i V ) (21) x i l i, i(c i = C, V i V ) (22) x iε{0, 1} (23) Z h, Z l, Z i 0 (24) NUMERICAL EXPERIMENT The propose moel is an integer programming problem. This section presents a verifie case with smaller problem scale for iscussion in etail. Then, the solution results for the real-worl case of 2015-2016 regular season tournaments are reporte. Verifie Case The test case only selects 12 teams, as shown in Table 1, for tournaments scheuling to play 30 games each team within 65 ays. Detaile settings of input ata for scheuling regulations are presente in Table 2. Table 1: Teams in the verifie case Team Location Winning Percentages (last season) Conference Division Celtics Boston 0.488 Eastern Atlantic Nets Brooklyn 0.463 Eastern Atlantic Bulls Chicago 0.610 Eastern Central Cavaliers Clevelan 0.646 Eastern Central Hawks Atlanta 0.732 Eastern Southeast Hornets Charlotte 0.402 Eastern Southeast Mavericks Dallas 0.610 Western Southwest Rockets Houston 0.683 Western Southwest Nuggets Denver 0.366 Western Northwest Timberwolves Minnesota 0.195 Western Northwest Clippers Los Angeles 0.683 Western Pacific Lakers Los Angeles 0.256 Western Pacific 22
Lu et al. / A tournment scheuling moel for Table 2: Settings of input ata for the test case Regulations Settings The total number of home (roa) games for same conference 2 same ivision The total number of home (roa) games for ifferent conferences 1 Upper boun of total number of home (roa) games for same 2 conference ifferent ivisions Lower boun of total number of home (roa) games for same 1 conference ifferent ivisions Lower boun of total number of home an roa games for same 3 conference ifferent ivisions The total ays of a season 65 The total number of games of each team 30 The total number of games on Friay an Saturay 4 Do not arrange games Day 20, 21, 22, 23, 24 To scheule special games on special ate Day 2 Celtics vs. Nets Day 14 Cavaliers vs. Lakers Day 14 Bulls vs. Clippers To scheule roa games for i team in a particular perio of time Clippers Day 10, 11, 12, 13, 14 Lakers Day 10, 11, 12, 13, 14 cannot use the home team ientity to play game on same ay Lakers an Clippers Each team only has 2 games at most in 3 ays This case was solve by calling the commercial optimization package CPLEX 12.4 on a computer with the platform of Winows 7 operating system an Intel(R) Core(TM) i5-4210u CPU @ 1.70GHz 2.40 GHz. The problem inclues 9,374 variables an 1,898 constraints. It expense 10 CPU secons to obtain the obective value of 1.525. This obective value is the ifference of highest opponent competitiveness, Hornets, an the lowest one of Clippers. The other results are shown in Figure 2. Figure 2. Test results for opponent competitiveness of each team in the verifie case Scheules for all of teams can follow the necessary regulations. Tables 3 an 4 illustrate the scheules of Clippers an Lakers. The number of plays against each opponent is between 2 an 4 games. Each team plays 30 games for the same home an away ones. The ates limite to play are also avoie to arrange any tournament. Conference an ivision matches are also followe. 23
Lu et al. / A tournment scheuling moel for Table 3: Test results for clippers scheule in the verifie case Day Home Team Roa Team Day Home Team Roa Team 2 We Clippers Rockets 32 Fri Clippers Nuggets 4 Fri Clippers Hawks 35 Mon Mavericks Clippers 5 Sat Hawks Clippers 36 Tue Clippers Nets 7 Mon Clippers Mavericks 38 Thu Rockets Clippers 9 We Mavericks Clippers 40 Sat Clippers Cavaliers 11 Fri Timberwolves Clippers 43 Tue Cavaliers Clippers 12 Sat Nuggets Clippers 48 Sun Clippers Hornets 14 Mon Bulls Clippers 50 Tue Clippers Timberwolves 16 We Clippers Mavericks 53 Fri Clippers Bulls 18 Fri Lakers Clippers 54 Sat Clippers Lakers 19 Sat Nets Clippers 57 Tue Lakers Clippers 26 Sat Clippers Timberwolves 59 Thu Celtics Clippers 28 Mon Clippers Lakers 62 Sun Nuggets Clippers 29 Tue Clippers Rockets 64 Tue Rockets Clippers 31 Thu Hornets Clippers 65 We Clippers Celtics Table 4: Test results for lakers scheule in the verifie case Day Home Team Roa Team Day Home Team Roa Team 1 Tue Timberwolves Lakers 40 Sat Timberwolves Lakers 5 Sat Bulls Lakers 42 Mon Nets Lakers 9 We Celtics Lakers 43 Tue Mavericks Lakers 14 Mon Cavaliers Lakers 46 Fri Lakers Nuggets 15 Tue Hornets Lakers 47 Sat Nuggets Lakers 17 Thu Lakers Celtics 49 Mon Lakers Timberwolves 18 Fri Lakers Clippers 51 We Nuggets Lakers 25 Fri Lakers Hornets 52 Thu Lakers Timberwolves 27 Sun Lakers Nets 54 Sat Clippers Lakers 28 Mon Clippers Lakers 56 Mon Lakers Mavericks 30 We Lakers Hawks 57 Tue Lakers Clippers 31 Thu Lakers Nuggets 60 Fri Lakers Rockets 33 Sat Rockets Lakers 61 Sat Lakers Mavericks 35 Mon Lakers Bulls 64 Tue Lakers Cavaliers 39 Fri Rockets Lakers 65 We Hawks Lakers Real-Worl Case The propose moel was also applie to solve the real-worl case for 2015-2016 regular season. This case has 153,032 variables an 11,866 constraints. The solution time is 2,174 CPU secons. The highest opponent competitiveness occurs on Timberwolves for 42.298, while the lowest opponent competitiveness appears on Hawks for 39.866. The gap of 2.432 between these two teams is smaller than the ifference of opponent competitiveness, 3.711, in 2015-16 season official scheule. The etaile results for opponent competitiveness are shown in Figures 3 an 4. The preliminary numerical experiments reveale that the propose moel can obtain promising results for the tournament arrangement of the NBA. 24
Lu et al. / A tournment scheuling moel for Figure 3. Opponent competitiveness of each team in solution results Figure 4. Opponent competitiveness of each team in 2015-16 season official scheule CONCLUSION AND SUGGESTIONS Professional sport tournaments are very fascinating. Every league has accumulate a lot of fans an then brought huge benefits. Tournament scheuling problem is an important part for running a professional sport system. This stuy has propose a mathematical moel to scheule the professional sports tournament scheule for the NBA with a fair perspective. The numerical experiments reveale that this moel can obtain promising results. This moel or the formulate perspective can also apply to other sports for tournament scheuling. Although this stuy took the winning percentages to be the obective, other obective concept, such as travel time, travel istance, back to back quantities, etc., can be also involve in the future research for the NBA. 25
Lu et al. / A tournment scheuling moel for REFERENCES Bean, J. C., an Birge, J. R. 1980. Reucing Travelling Costs an Player Fatigue in the National Basketball Association. Interfaces 10(3): 98-102. Costa, D. 1995. An Evolutionary Tabu Search Algorithm an the NHL Scheuling Problem. INFOR: Information Systems an Operational Research 33(3): 161-178. Dilkina, B. N., an Havens, W. S. 2004. The National Football League Scheuling Problem. Presente at the Innovative Applications of Artificial Intelligence Conference (IAAI-04), San Jose, CA. Duran, G., Guaaro, M., an Wolf-Yalin, R. 2012. Operations Research Techniques for Scheuling Chile s Secon Division Soccer League. Interfaces 42(3): 273-285. Garcia, J., Ibanez, S. J., Martinez De Santos, R., Leite, N., an Sampaio, J. 2013. Ientifying Basketball Performance Inicators in Regular Season an Playoff Games. Journal of Human Kinetics 36(1): 161-168. Nemhauser, G. L., an Trick, M. A. 1998. Scheuling a Maor College Basketball Conference. Operations Research 46(1): 1-8. Polog, L., Buhler, C. F., Pollack, H., Hopkins, P. N., an Burgess, P. R. 2015. Time Trens for Inuries an Illness, an Their Relation to Performance in the National Basketball Association. Journal of Science an Meicine in Sport 18(3): 278-282. Ribeiro, C. C. 2012. Sports Scheuling: Problems an Applications. International Transactions in Operational Research 19(1-2): 201-226. Summers, M. R. 2013. How to Win in the NBA Playoffs: A Statistical Analysis. American Journal of Management 13(3): 11-24. Wright, M. B. 2006. Scheuling Fixtures for Basketball New Zealan. Computers & Operations Research 33(7): 1875-1893. Wright, M. B. 2009. 50 Years of OR in Sport. Journal of the Operational Research Society 60(1): S161-S168. 26