Scheduling the Chilean Soccer League by Integer Programmingmate.dm.uba.ar/~gduran/docs/papers/OR_Chilean_soccer.pdf · 2008. 6. 7. · Duránetal.:Scheduling the Chilean Soccer League

Vol. 37, No. 6, November–December 2007, pp. 539–552issn 0092-2102 �eissn 1526-551X �07 �3706 �0539

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doi 10.1287/inte.1070.0318©2007 INFORMS

Scheduling the Chilean Soccer League byInteger Programming

Guillermo Durán, Mario Guajardo, Jaime Miranda, Denis Sauré,Sebastián Souyris, Andres Weintraub, Rodrigo Wolf

Department of Industrial Engineering, University of Chile, Santiago, Chile{[email protected], [email protected], [email protected], [email protected],

[email protected], [email protected], [email protected]}

Since 2005, Chile’s professional soccer league has used a game-scheduling system that is based on an integer lin-ear programming model. The Chilean league managers considered several operational, economic, and sportingcriteria for the final tournaments’ scheduling. Thus, they created a highly constrained problem that had been,in practice, unsolvable using their previous methodology. This led to the adoption of a model that used sometechniques that were new in soccer-league sports scheduling. The schedules they generated provided the teamswith benefits such as lower costs, higher incomes, and fairer seasons. In addition, the tournaments were moreattractive to sports fans. The success of the new scheduling system has completely fulfilled the expectations ofthe Asociación Nacional de Fútbol Profesional (ANFP), the organization for Chilean professional soccer.

Key words : Chilean soccer league; integer programming; sports scheduling; recreation/sports; OR/MSimplementation; scheduling.

History : This paper was refereed.

Soccer is “the passion of multitudes” worldwide.The World Cup 2006, which was held in Germany,

amply demonstrated this phenomenon. Beyond thepurely sporting and emotional aspects of the game,managing soccer increasingly requires the applicationof scientific criteria. In Chile, soccer has faced compe-tition from the international leagues, other televisedsports, and different forms of entertainment such asshopping malls, cinema, video games, and the Inter-net. Organizers of professional sports in many coun-tries are facing similar situations.

This competition has caused a reduction inChileans’ interest in soccer and a resulting decline inthe revenue the sport generates. To reverse this declineand reduce costs, professional league officials are fac-ing the challenge of having to increase the attractive-ness of the league season. The planning of leaguegame schedules is critical to achieving this objec-tive. Considering the many aspects of developing agame calendar that is simultaneously fair to the teams,economically beneficial, and attractive to sports fans,the task of scheduling each regular-season matchupwould be nearly impossible if attempted manually.

Beginning with the first tournament of 2005, theAsociación Nacional de Fútbol Profesional (ANFP),the organizing body for Chilean soccer, employedthe services of the Centro de Gestión de Opera-ciones (CGO), a unit of the Industrial EngineeringDepartment at the University of Chile, to assist inthe planning of the main league’s game schedule. Weintegrated sporting, operational, and economic crite-ria within an integer programming model to developa schedule that meets the ANFP criteria and makesthe season more interesting to soccer fans.

Our work falls within the area known as sportsscheduling. In this paper, we present the criteriawe used to define an efficient season schedule. Weaddress sporting fairness or equity, the introductionof operational and economic considerations into thescheduling process, and how the model and its imple-mentation provide a flexibility that was previouslyabsent in Chilean soccer schedules. In addition toincreasing the season’s attractiveness, these factorscombine to put the scheduling process on a more sci-entific basis, making it more transparent and moreacceptable to team managers.

539

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Durán et al.: Scheduling the Chilean Soccer League by Integer Programming540 Interfaces 37(6), pp. 539–552, © 2007 INFORMS

We organize the paper as follows. We begin witha description of the Chilean soccer tournaments anda review of the sports-scheduling literature. In theConditions Imposed on the Problem section, we explainthe conditions considered in the 2006 opening tour-nament. In The Mathematical Model and ComputationalSolution section, we describe the model and its com-putational solution. In the Results section, we discusssome recent statistics and qualitative factors of ourmodel that have satisfied the ANFP and soccer teamsand fans. In the Conclusions section, we summarizeand provide guidelines for future work. Finally, in theappendix, we show the formulation of the mathemat-ical model.

BackgroundThe Chilean professional soccer league is composed ofthe First Division and the Second Division. The FirstDivision has 20 teams and divides its annual playingcalendar into two tournaments—the opening cham-pionship and the closing championship. Each cham-pionship comprises two phases: the regular season,which consists of 19 playing dates or rounds, and theplayoffs. The teams are organized into four groups offive teams each. Each team must play once againsteach of the other 19 teams. The ANFP sets the dateof each round and the composition of the groups inadvance. Following the regular season, the two topteams in each group advance to the playoffs to deter-mine the champion. The Mexican soccer-league sys-tem inspired this structure.

The Second Division has 12 teams. Each year thelast two teams (measured by the sum of points in bothtournaments) of the First Division are relegated to theSecond Division; the best two teams of the SecondDivision are promoted to the First Division.

The country is geographically divided into 12Regions and the Metropolitan Region (Santiago),which is located between Regions V and VI. Wehave classified the 20 teams of the First Division intothree clusters by geographic location: North with fiveteams, Center with 10 teams, and South with fiveteams (Figure 1).

Prior to 2005, the method used to schedule the FirstDivision was a random draw of teams and venuesusing a preset template; almost all soccer leagues in

South America and Europe use this system. Whileit facilitated manual scheduling, it did not take intoaccount most of the criteria (e.g., fairness and eco-nomic considerations to provide greater revenue andlower costs for the teams) that are needed for effi-ciency. To ensure fairness, each team should playa balanced mix of home and away games againstthe strongest teams. Games against the strongestteams should not be scheduled consecutively; eachteam should play against two of its group oppo-nents at home and against the other two oppo-nents away. Scheduling two consecutive away games(e.g., Sunday–Wednesday or Wednesday–Sunday) fora given team in different opponents’ venues, whichare located relatively close to each other but far fromthe team’s home venue, is an example of an economicconsideration; it would spare the team a secondlong trip. Other examples include setting “attractive”games for appropriate dates, such as summer homegames for teams located in popular beach townsagainst the most popular teams, and scheduling clas-sic rivalries or matches between teams of the samegroup in the second half of the tournament when thestakes are higher. In addition, we can distribute week-day home games fairly; such dates are less attrac-tive to the teams because attendance is lower thanon weekends (revenue for any specific game goesentirely to the home team).

In previous years, scheduling based on these cri-teria was woefully deficient. Examples included clas-sic matchups on inappropriate dates, weaker teamsscheduled to play all their games against strongerones away from home, and unbalanced distributionof weekday, home games.

Because a season calendar that meets these stan-dards of efficiency would be nearly impossible todevelop manually, operations research can make asubstantial contribution by allowing the applicationof technology to give more flexibility to the season-scheduling process. To help the reader appreciate thescale of the complexity involved, we note that for atournament in which six teams play a simple round-robin, there are 720 different possible schedules (thisdoes not even consider whether the games are athome or away); for a tournament with eight teams,there are more than 30 million possibilities. Clearly,the number of possible schedules for the 20 soccer

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Durán et al.: Scheduling the Chilean Soccer League by Integer ProgrammingInterfaces 37(6), pp. 539–552, © 2007 INFORMS 541

Region I

Region II

Region III

Region IV

Region VI

Region VII

Region VIIIRegion IX

Region XI

Region XII

Region X

Region V

Santiago

N

S

EW

NORTH

ANTF

CBLOA

CBSAL

LSRN

CQMB

CENTER

EVRT

WDRS

UCH

UE

COLO

CATO

PLTN

STGM

AUDAX

OHG

SOUTH

RNGS

UDC

CONCE

HCH

PMNTT

Figure 1: The map of Chile shows the locations of the 20 teams of the First Division in the opening tournamentin 2006.

teams in Chile’s First Division would be too enor-mous to imagine.

The use of sports-scheduling techniques is still anovelty in South America. The only documented casewas E. Dubuc’s development of such a method for theArgentinean soccer league in the 1995 season (Paenza2006), but the practice was subsequently abandoned.Bartsch et al. (2006) have also reported isolated casesin European soccer. In the United States, on the otherhand, sports scheduling is routinely employed by mostmajor basketball, baseball, and football leagues thatmaintain their own teams of academics or hire third-party companies to design efficient regular-seasonschedules (Bean and Birge 1980, Nemhauser and Trick1998, Henz 2001, Associated Press 2004).

At the academic level, the literature on sportsscheduling has grown significantly in recent years.Various articles have been published proposing as yetuntried scheduling applications for existing leagues(Costa 1995, Della Croce and Oliveri 2006, Schreuder1992). Interest in the problem increased notably withthe publication of the traveling tournament prob-lem (TTP) (Easton et al. 2001), which addresses thedesign of a schedule to minimize the distances thatteams in a sports league must travel. Although it isnot set up as a real case, the TTP has provided asignificant benchmark for using a range of methodsand algorithms. Anagnostopoulos et al. (2006) pro-pose a heuristic for solving the TTP based on sim-ulated annealing, while Cardemil and Durán (2004)

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present a tabu-search application with the same objec-tive. Easton et al. (2003) offer a combination of inte-ger programming and constraint programming as amethod of finding the optimal solution for leagues ofup to eight teams. Ribeiro and Urrutia (2006) developheuristics for the mirrored version of the TTP (doubleround-robin tournament in which the game order ineach single round-robin is the same).

As we will describe later, the goal of the Chileantournament is not to minimize travel distances pre-cisely, but rather to develop a schedule that satisfiesa long and complex list of conditions, which togethercreate a highly constrained problem.

Conditions Imposed on the ProblemThe schedule must meet specific conditions to complywith the requirements that the ANFP established forthe 2006 opening tournament. Most criteria were alsoamong those applied to the two 2005 tournaments.The appendix shows the mathematical formulationsfor these conditions.

Basic Schedule Constraints(1) Each team must play each of the other teams

once during the 19 tournament rounds.(2) Each team must play each round either at home

or away.(3) Each team must play at least nine rounds, but

not more than 10, at home.

Home and Away Game Sequence Constraints(4) Each team must play at most one sequence of

two consecutive rounds at home (home stand). Notethat this condition also implies that no team playsmore than two consecutive rounds at home.

(5) Each team must play at most one sequence oftwo consecutive rounds away (road trip). Note thatthis condition also implies that no team plays morethan two consecutive rounds away.

(6) Let A be a set of rounds such that if a teamplays at home (away) in a round belonging to A, itmust play away (at home) in the following round. Inthe 2006 opening game, the set was defined as A =�1�16�18�. The objective is to balance the teams’ homeand away games between the early and late stages ofthe tournament.

Home Game Balance Constraint for MatchesAgainst Group Rivals

(7) Each team must play against two of its groupopponents at home and against the other two oppo-nents away.

Geographic Constraints for Double Away GameSequencesWe have incorporated certain constraints to avoidconsecutive long road trips.

(8) When a North (South) team plays two consec-utive away games, neither of these games will beplayed in the South (North).

(9) When a North (South) team plays two consec-utive away games, at least one of the games will beplayed in the North (South).

(10) When a Center team plays two consecutiveaway games, at least one of the games will be playedin the Center.

It is important to note that in the 2006 opening tour-nament, all the rounds were played on weekends. (Wenote that it is very expensive for a team to stay awayfrom its city for an entire week.)

Therefore, we could not schedule good tripsin consecutive away games Sunday–Wednesday orWednesday–Sunday.

Constraints on Highly Popular TeamsIn our model, we placed constraints on the three mostpopular teams—Colo Colo, Universidad Católica, andUniversidad de Chile. In the appendix, figures, andtables, we refer to these three teams as COLO, CATO,and UCH, respectively.

(11) If a team plays at home (away) against ColoColo, it plays away (at home) against Universidad deChile. This contributes both to fairness and a betterbalance of revenue between the opening and closingtournaments because games against these teams usu-ally generate higher receipts.

(12) The three classic matchups between these pop-ular teams must be played between the 10th and 16throunds; however, parameters may change from tour-nament to tournament.

(13) Each of these three teams plays exactly oneclassic matchup at home.

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Operational Constraints on the Availability ofMobile Broadcasting Units for Televising GamesFour games from each round are televised. Theseinclude all of the games that involve the three pop-ular teams plus a fourth match. The two opposingteams that have the highest combined point total playthe fourth match; if there is a classic game in a givenround, two games are selected in this way. GivenChile’s large geographical area, it is desirable thatthese teams not play in venues that are located farapart; this limits travel distances and the associatedcosts that the television broadcasting company mustincur for the transfer of mobile units to and from thegames.

(14) When a popular team plays in the North(South), neither of the two other popular teams canplay in the South (North).

(15) Given that the first five rounds are scheduledfor the middle of the summer (when many events aretelevised and the availability of mobile units is lower),none of the three popular teams play away in thoserounds in outlying areas of the country. We defineoutlying areas as the members of a set containing theteams whose home venues are located north of Chile’sRegion IV or south of Region VIII.

Constraints on Strong Teams, Defined as the ThreePopular Teams Plus Cobreloa (CBLOA), Which Is aFourth Strong Team

(16) No team may play two consecutive gamesagainst a strong team.

(17) Games between CBLOA and the popularteams are played between the 6th and 18th rounds.

Constraints on Home and Away Games for“Crossed” TeamsWe refer to a pair of teams as crossed if their homevenues are in the same region and if they alternatewith each other playing home and away games ineach round; they may alternate for operational rea-sons (e.g., they share the same home stadium), secu-rity reasons, or in order not to leave any regionwithout a match for an entire weekend. In total, therewere five crossed pairs: Wanderers (WDRS) and Ever-ton (EVRT), both of which are Region V teams, andColo Colo and Universidad de Chile, which are themost important teams in the country and are locatedin Santiago, are examples of crossed pairs.

(18) When one team of a crossed pair is playing athome, the other team plays away, and vice versa.

Constraints on Regional Classic MatchupsWe define a number of pairs of teams from the sameregion and with a historic rivalry as a set of regionalclassics.

(19) Regional classic matchups are held betweenthe 8th and 18th rounds.

Constraints on Santiago Games(20) The number of games held in Santiago in each

round cannot be less than two or more than four(there are seven Santiago teams). This enables the soc-cer activity in the capital to be regulated and ensuresthe availability of stadiums and municipal securitypersonnel.

(21) It is desirable that the four Santiago teams withthe lowest drawing power not play against each otherin the first five rounds (all are in the summer) becausethe attendance would be relatively low. We define aset D that consists of these four teams.

Tourism-Related ConstraintsWe define a set T that contains teams located intourist areas, where it is desirable that at least oneattractive game (against a popular team) be sched-uled for the first rounds of the tournament during thesummer.

(22) Each team located in a tourist area plays athome against at least one of the popular teams in oneof the first five rounds.

Special Constraints(23) Some teams do not have their playing fields

ready in time for the beginning of the tournament;therefore, they should be scheduled for away gamesin the first round. These teams form a set S.

(24) Not more than three games between teams ofthe same group are held in the last round. In 2006, theANFP made this condition a requirement because inthe past tournaments, the playoff qualifiers for mostof the groups were defined before the very last round.

(25) Each North (South) team shall play at leastonce at home against a North (South) team. This con-straint will ensure that, in the 2006 closing tourna-ment, North and South teams play against at leastone opponent in their respective clusters. This could

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help to avoid “bad” two-game road trips. (Note thatfor Center teams, the same condition is guaranteed byconstraints 11 and 13 because the three highly popularteams are from the Center cluster.)

We have also incorporated special constraints inthe other tournaments we have scheduled. Examplesinclude not scheduling home games for a team ondates when its stadium is booked for other events andavoiding road trips for teams on dates that are closeto dates when they must travel to an international cupgame. We also scheduled, in the first round of a tour-nament, a match between teams that are playing inthe playoff finals of the immediately preceding cham-pionship. Therefore, this game could be rescheduledat a later date to allow these two teams an additionalweek of rest. Otherwise, the ANFP would have toreschedule two games.

Because the 2005 tournaments included weekdaygames, we took into account other factors that relateto fairness or economic considerations. In the 2005opening, which included only one round played onWednesday, the ANFP requested that a “good” roadtrip be scheduled for at least three teams between thatround and an immediately adjacent one. The closinghad three Wednesday rounds in the schedule; eachteam was required to play at home on at most twoof these rounds (clearly, in these two tournaments,constraints 8, 9, and 10 were not considered for con-secutive rounds including weekday games). Finally,for each of the championships, we formulated anobjective function that maximized the concentrationof “decisive” games among the final rounds of theschedule. In the opening tournaments of 2005 and2006, decisive games were defined as those betweenteams in the same group. In the 2005 closing, this def-inition was broadened to include the games betweenteams that were expected, based on their performancein the opening tournament, to be fighting against rel-egation to the Second Division. The appendix showsthe detailed formulation of the objective function.

We note the importance of the iterative process(by the group of academic experts and the ANFP) inarriving at a definitive schedule. Three or four dif-ferent versions of the schedule may bounce back andforth between them as they fine-tune various detailsto incorporate new constraints into the model beforeputting a set of final proposals forward. This process

begins one month before the publication of the defini-tive schedule.

The Mathematical Model and theComputational SolutionWe expressed the conditions described in the previ-ous section in terms of an integer linear programmingmodel (appendix). After we built the model, we usedCPLEX 9.0 on a Pentium IV computer with a 2.4 GHzprocessor to solve it.

The problem we face is essentially one of feasibility.Our main goal is to find a schedule that meets all ofthe conditions that we imposed on it. The objectivefunction simply measures how much we can “push”the decisive games toward the end of the tournament.In our situation, it is not as important that we arriveat an optimal solution as it is in most other optimiza-tion problems. The ANFP selects the final schedulefrom among a series of options that we present to it(following the iterative process discussed above); thechoice it makes is not necessarily the one that per-forms best according to the objective function.

The principal family of decision variables in themodel, which we describe in the appendix, takes ona value of one when team i plays at home againstteam j in round k, and zero otherwise. Other variablesexpress some of the more complex conditions. Themodel as formulated is extremely hard to solve. Tosimplify the solution, we incorporated an additionalfactor that establishes certain home-away patterns forthe teams and adds some rigidity to the formulation.

Ahome-awaypattern (Easton et al. 2003,NemhauserandTrick 1998) is a sequence assigned to a given teamthat indicates the number of rounds it will play athome and away. A sequence consisting of the 19 ele-ments �H�A� �H� represents a pattern in which Hsignifies a home-game round and A an away-gameround. The appendix shows the constraints that weapplied to the assignment of home-away patterns.The advantage of this approach with home-away pat-terns is a solution-time reduction that is achievedwithout significant loss of solution quality. Further-more, it can ensure a priori that certain constraintsthat normally account for the most serious prolonga-tions of model run time are satisfied. Some of theseconstraints regard crossed pairs of teams and theavoidance of double home stands or road trips.

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For the 2005 opening tournament, we found a feasi-ble solution using constraint programming (CP) afterapproximately three hours of run time. The model wesolved was similar to the one that we show in theappendix; however, it is simply a feasibility problemand is without the objective function. We used ILOG’sSolver 5.2 software and CP because of its good per-formance history in solving similar sports-schedulingproblems (Henz 2001).

Once we found this feasible solution, we assignedthe set of patterns we obtained to each team. We exe-cuted the optimization process and obtained the opti-mum solution for this pattern assignment in minutes.The schedule approved for this tournament featured24 of the 40 games between teams of the same groupin the last three rounds, much to the ANFP’s satisfac-tion. Note that this solution is the maximum possiblenumber for three rounds because at most eight gamesbetween teams of the same group can be played in agiven round.

For the 2005 closing tournament, we adopted a sim-ilar procedure; however, we partially incorporated theassignment of “feasible” patterns into the optimiza-tion process. To keep the solution time to a reasonablelevel, we allowed the process to assign some patternsand assigned others as fixed. We discovered throughexperimentation that with up to about 15 fixed pat-terns and five assigned by the process, we still arrivedat the solution quickly; it required no more than15 minutes, and was better than the solution obtainedusing the method employed for the opening. Thus,the calendar approved for the 2005 closing scheduledthe great majority of games between teams from thesame group and between teams that were expectedto be fighting one other to avoid relegation to theSecond Division in the second half of the tourna-ment. A closing tournament is significantly more con-strained because the home condition for a given pairof teams is already defined as the opposite of theone scheduled for the opening tournament (i.e., ifteam i played at home against team j in the opening,then team j must play at home against team i in theclosing).

For the 2006 opening tournament, we also sought afeasible solution as a first step. This involved impos-ing a set of patterns that we derived from the con-tents of the definitive solution for the 2005 closing, but

with modifications that would eliminate all doublehome stands and road trips and impose several dif-ferent sequences for rounds in which a concentrationof intergroup games was desired. Once we imposedthese patterns, we found a feasible solution in min-utes. We then took the set of patterns in this solutionand, rather than assigning them to the teams as is, wecompletely incorporated them into the optimizationprocess based on the solution of the relaxed problem.We solved the LP problem (i.e., without the constraintof integrality of the variables) and then set the “pat-tern” variable zip with the highest value in the LPsolution to 1. We repeated this procedure sequentiallyto arrive at a feasible solution. When the model didnot generate a feasible result, we backtracked throughthe iterations and changed the pattern assignmentin a logical way. This heuristic guided us quickly tosolutions that were better than our original solutionbecause the LP problem can be solved in seconds.Once the 20 patterns were fixed, we proceeded as wehad with the 2005 closing, applying this solution tothe model as the initial solution while allowing a fewpatterns (between two and five) to be reassigned inan attempt to further improve the objective functionvalue. Figure 2 shows the solution that ANFP selectedfor the 2006 opening tournament.

In this schedule, 100 percent of the games betweenteams in a given group were set for the 10th round orlater rounds, and slightly more than 80 percent wereconcentrated between the 14th and the 19th rounds—the rounds most highly weighted in the objectivefunction. The value of the feasible solution in theobjective function was 607, with a gap of 5.6 percentseparating it from the relaxed problem solution (tak-ing into account the last cut that the appendix shows).

ResultsIt is not easy to measure the impact of using the sys-tem we describe because there are many factors otherthan scheduling that influence variables, e.g., atten-dance at stadiums. Nevertheless, we can make cer-tain observations. In both 2004 tournaments, the lastunder the old system, the classic matchup betweenUniversidad de Chile and Colo Colo was held in thefirst round and drew crowds of 26,000 people in theopening and 22,000 in the closing. By contrast, in

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Teams\Match 1 2 3 4 5 6 7 8 9

UCH UE @RNGS UDC @WDRS AUDAX @CBSAL PMNTT @ANTF PLTNCOLO @OHG HCH @LSRN EVRT @PLTN WDRS @CONCE CBSAL @STGMCATO @EVRT PMNTT UE @CQMB ANTF @CBLOA @PLTN LSRN @OHGANTF @HCH PLTN @EVRT STGM @CATO OHG @LSRN UCH @CONCECBLOA @CBSAL CQMB @STGM CONCE @OHG CATO @HCH RNGS @WDRSCBSAL CBLOA @LSRN @OHG PMNTT @STGM UCH EVRT @COLO CQMBLSRN @CONCE CBSAL COLO @RNGS UE @AUDAX ANTF @CATO HCHCQMB UDC @CBLOA @PLTN CATO @PMNTT HCH @WDRS OHG @CBSALEVRT CATO @AUDAX ANTF @COLO @HCH CONCE @CBSAL PLTN @PMNTTWDRS @PLTN OHG @CONCE UCH RNGS @COLO CQMB @UE CBLOAAUDAX @PMNTT EVRT @HCH OHG @UCH LSRN @STGM CONCE @RNGSUE @UCH CONCE @CATO HCH @LSRN PLTN @RNGS WDRS UDCPLTN WDRS @ANTF CQMB @UDC COLO @UE CATO @EVRT @UCHSTGM RNGS @UDC CBLOA @ANTF CBSAL @PMNTT AUDAX @HCH COLOOHG COLO @WDRS CBSAL @AUDAX CBLOA @ANTF UDC @CQMB CATORNGS @STGM UCH @PMNTT LSRN @WDRS @UDC UE @CBLOA AUDAXUDC @CQMB STGM @UCH PLTN @CONCE RNGS @OHG PMNTT @UECONCE LSRN @UE WDRS @CBLOA UDC @EVRT COLO @AUDAX ANTFHCH ANTF @COLO AUDAX @UE EVRT @CQMB CBLOA STGM @LSRNPMNTT AUDAX @CATO RNGS @CBSAL CQMB STGM @UCH @UDC EVRT

Teams\Match 10 11 12 13 14 15 16 17 18 19

UCH @HCH COLO OHG @CQMB STGM @EVRT @CATO LSRN @CBLOA CONCECOLO RNGS @UCH @AUDAX CATO @UE CBLOA CQMB @PMNTT ANTF @UDCCATO CONCE @STGM RNGS @COLO WDRS @AUDAX UCH @UDC CBSAL @HCHANTF WDRS @RNGS CBSAL @UDC AUDAX @PMNTT CBLOA @CQMB @COLO UECBLOA LSRN @UDC UE @PLTN PMNTT @COLO @ANTF AUDAX UCH @EVRTCBSAL @UDC PLTN @ANTF HCH @RNGS WDRS @UE CONCE @CATO AUDAXLSRN @CBLOA @OHG WDRS @PMNTT UDC @STGM EVRT @UCH CQMB @PLTNCQMB STGM AUDAX @CONCE UCH @EVRT UE @COLO ANTF @LSRN RNGSEVRT OHG @UE UDC @STGM CQMB UCH @LSRN WDRS @RNGS CBLOAWDRS @ANTF HCH @LSRN AUDAX @CATO @CBSAL UDC @EVRT STGM @PMNTTAUDAX UE @CQMB COLO @WDRS @ANTF CATO PLTN @CBLOA UDC @CBSALUE @AUDAX EVRT @CBLOA @OHG COLO @CQMB CBSAL @STGM PMNTT @ANTFPLTN PMNTT @CBSAL STGM CBLOA @HCH CONCE @AUDAX RNGS @OHG LSRNSTGM @CQMB CATO @PLTN EVRT @UCH LSRN @CONCE UE @WDRS OHGOHG @EVRT LSRN @UCH UE @CONCE RNGS PMNTT @HCH PLTN @STGMRNGS @COLO ANTF @CATO CONCE CBSAL @OHG HCH @PLTN EVRT @CQMBUDC CBSAL CBLOA @EVRT ANTF @LSRN HCH @WDRS CATO @AUDAX COLOCONCE @CATO @PMNTT CQMB @RNGS OHG @PLTN STGM @CBSAL HCH @UCHHCH UCH @WDRS PMNTT @CBSAL PLTN @UDC @RNGS OHG @CONCE CATOPMNTT @PLTN CONCE @HCH LSRN @CBLOA ANTF @OHG COLO @UE WDRS

Figure 2: The schedule approved for the 2006 opening tournament shows games between teams of the samegroup in grey. Note that all of them are scheduled for the second half of the championship.

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Average attendance and ticket revenue per game

2004 opening 2004 closing 2006 opening

Attendance 3�756 3�557 4�953Ticket revenue 5�852 6�095 11�803

Table 1: Comparisons of average game attendance and ticket revenue (inthousands of Chilean pesos) in the 2004 opening and closing tournamentsand the 2006 opening tournament. Figures refer only to regular seasongames. (Data supplied by ANFP.)

the two 2005 tournaments and the 2006 opening, thismatchup was scheduled further into the season andthe attendance jumped to 45,000, 37,000, and 49,000,respectively.

Two additional indicators of interest are the atten-dance and revenue averages, both measured pergame, during the regular season (the new schedulingcriteria do not affect the playoffs). Table 1 summa-rizes the data for the last two tournaments using theold scheduling system (i.e., 2004 opening and clos-ing) and the latest tournament played (2006 opening).They reveal that the average rise in attendance was32 percent when compared to the 2004 opening and39 percent over that year’s closing; the correspond-ing average increases for ticket sales were 102 and 94percent, respectively.

Table 2 compares attendance and ticket revenue forthe classic games of the 2004 tournaments (still usingthe old scheduling method) with those of the 2005 and2006 tournaments scheduled using the new system.

When the same team was at home, all of the2005 matchups drew better crowds and generatedmore revenue than in 2004; total attendance increased74 percent and receipts increased 142 percent. The2006 opening classics (once again, with the same teamat home) drew 124 percent more than their 2004equivalents. This translated into a revenue increase of347 percent.

Table 3 displays comparisons for home gamesplayed in summer rounds by teams in Region IV—Coquimbo (CQMB) and La Serena (LSRN)—andRegion V—Everton (EVRT) and Wanderers (WDRS)—against the popular teams.

According to the data, attendance grew by 46 per-cent for Region IV matches between 2004, when theywere not played in summer, and 2006, when they

Attendance and ticket revenue for classic games

UCH @ COLO COLO @ UCH

2004 op. 2005 op. 2004 cl. 2005 cl. 2006 op.

Date Feb-08-04 Apr-10-05 Aug-01-04 Aug-28-05 Apr-09-06Attendance 25�743 45�236 21�750 37�420 48�996Ticket 55�900 114�879 59�967 137�394 240�557revenue

CATO @ UCH UCH @ CATO

2004 cl. 2005 op. 2004 op. 2005 cl. 2006 op.

Date Nov-04-04 Apr-30-05 May-04-04 Nov-13-05 May-14-06Attendance 18�093 24�450 7�881 18�292 14�409Ticket 55�173 71�499 12�241 69�099 67�466revenue

COLO @ CATO CATO @ COLO

2004 op. 2005 op. 2004 cl. 2005 cl. 2006 op.

Date Mar-06-04 Mar-20-05 Sep-12-04 Sep-25-05 Apr-23-06Attendance 9�887 24�352 13�333 18�138 32�654Ticket 16�575 100�408 29�595 61�074 147�442revenue

Table 2: Comparisons of attendance and ticket revenue (in thousands ofChilean pesos) for classic games in 2004, 2005, and 2006. Figures refer tomatches in which the home team was the same. (Data supplied by ANFP.)

were. The improvement in Region V was particu-larly impressive at 156 percent. In both cases, revenuealso increased significantly—by 84 and 313 percent,respectively.

Attendance and ticket revenue for games in tourist areas

COLO @ LSRN CATO @ CQMB

Region IV 2004 2006 op. 2004 2006 op.

Date May-09-04 Feb-11-06 Mar-13-04 Feb-18-06Attendance 5�373 7�533 4�673 7�178Ticket revenue 10�175 22�101 10�681 16�175

CATO @ EVRT UCH @ WDRS

Region V 2004 2006 op. 2004 2006 op.

Date Apr-21-04 Jan-29-06 Oct-29-04 Feb-19-06Attendance 3�314 6�638 3�494 10�787Ticket revenue 3�840 13�857 6�563 29�091

Table 3: Comparisons of attendance and ticket revenue (in thousands ofChilean pesos) for games held in tourist areas in 2004, 2005, and 2006.Figures refer to matches in which the home team was the same in bothyears. (Data supplied by ANFP.)

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Furthermore, the adoption of criteria that reducethe costs of broadcasting games has been highly cele-brated by the television-station management and theANFP, which has the 80 percent share of the soc-cer television company. For example, two televisedgames played in the north may result in savings forthe television company of approximately US$20,000over the cost of broadcasting one game held in thenorth and the other in the south. In the future, it maybe possible to pass along some of these savings to theteams. The income derived from television rights isan important source of revenue for the Chilean teams.In the last negotiation, the league sold its televisionrights for approximately US$3,700,000. Forty percentof this amount was assigned to the three most popu-lar teams.

Although we may partly explain the quantitativeinformation in the preceding paragraphs by the newschedules, it is difficult to control for some exogenousfactors that are present. These factors, which includethe performance of the national team in World Cupqualifying rounds, weather patterns, acts of violencecommitted by team supporters, and the quality of theteams (fundamentally those that draw larger crowds)may distort the measurements.

The positive impact of the sporting-fairness crite-ria that we adopted in our model deserves specialmention. Both the ANFP and the league teams haveexpressed their satisfaction with how these criteriahave been put into practice. Senior ANFP officials alsonoted that, for the first time in many years, they havereceived almost no complaints from the teams regard-ing the schedules.

ConclusionsTo improve the solution process, the procedures wedescribed above for finding good solutions raise cer-tain issues that remain to be explored in future work.These include:

• Formalization of the solution to the integer prob-lem starting from the relaxed version via a heuris-tic that sequentially structures the fixing of the zippattern variables at one, thus detecting and repairinginfeasibilities.

• Incorporation of the creation of patterns in theoptimization process or use of a set of patterns of

higher cardinality than the number of teams. Thiscould lead to better solutions and greater flexibilityin the search for feasible solutions; however, the pricewill likely be increased solution times.

• Experimentation to determine how much thecuts shown in the appendix contribute in terms ofreducing solution time (recall that as with every inte-ger programming problem, the use of cuts to adjustthe feasible polyhedron of the linear relaxation maybe very useful).

The incorporation of these modern techniquesinto the Chilean soccer-league scheduling processsince 2005 has provided an excellent opportunity todemonstrate that the use of operations research canbe effective in making soccer-season schedules moreattractive to the public as well as fairer and moreprofitable for the teams and organizing bodies. It isalso important to highlight the transparency that themodel we discussed has brought to the system. Oncethe constraints to be applied have been defined andmade known to all concerned, they are incorporatedas part of the mathematical model. Then, we can gen-erate some possible solutions and allow the ANFPto select the definitive solution. This procedure alsorequires that the ANFP submit its objectives for theschedule to the teams; this facilitates consensus build-ing and the creation of new mechanisms for improv-ing the league’s scheduling process.

Thus far, we have scheduled the last five tour-naments of the First Division (2005 opening, 2005closing, 2006 opening, 2006 closing, and 2007 open-ing), the annual 2007 tournament of the Second Divi-sion, and the Chilean proposal for the scheduling ofSouth American qualifiers to the South Africa 2010World Cup.

There are several anecdotes that clearly illustratethe impact this scheduling application has had inChilean soccer circles; one is particularly revealing.The day that the 2005 closing-tournament sched-ule was publicized, the ANFP’s operational managernoted that he had received a complaint from thepresident of the Coquimbo team regarding its homegame against Colo Colo; the game had been sched-uled on the “Fiesta de la Pampilla,” a local holiday.This meant that the necessary police presence for thematch would not be available and that the stadiumattendance would be significantly reduced. Within

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a few hours, the game was switched to a differentround. In addition, because the overall schedule hadalready been published, the number of other changeswas minimized so that the modified version was assimilar as possible to the original schedule. The abil-ity to rely on a rapid and agile tool that would enableus to solve the problem in minimal time was crucialto the task.

Participants have also expressed other significantreactions to us. A player for Palestino, a Santiagoteam, commented that he was surprised by the“chance occurrence” that his team was scheduled toplay against its four group rivals in the last fiverounds of the 2005 opening, thus leading to more sig-nificant games. During the 2006 opening, a televisioncompany executive remarked on the “luck” of televis-ing games involving Universidad de Chile on a Sat-urday and Universidad Católica the next day whilethe two teams were on road trips in the north. Thissaved his company a significant amount of moneybecause of the proximity of the two venues and thecorrespondingly low mobile-unit transfer costs.

In conclusion, we observe that management tech-niques can make other contributions to SouthAmerica’s most popular sport. Issues such as resourcemanagement, management of the creation of lowerdivisions, new tournament formats, optimal ticketprices, strategic alliances with other countries inthe region, policies for encouraging the return oftop players currently playing abroad, and the effi-cient administration of the economic and operationalaspects of teams and related organizations, are someof the areas that could benefit from a more scientificand quantitative approach.

Appendix: Formulation of theMathematical ModelThe integer linear programming model used to gener-ate the 2006 Opening Tournament is described below.

VariablesTo define the games to be held in each round, wedefine ∀ i �= j ∈ I (the set of teams) and ∀k ∈K (the set

of rounds), a family of binary variables, as follows:

xijk =

1 if team i plays at home against

team j in round k,

0 otherwise.

To represent simply certain home and away gamesequence constraints, we define ∀ i ∈ I and ∀k =1� �18, the following auxiliary variables, alsobinary:

yik =

1 if team i plays at home in rounds

k and k+ 1,

0 otherwise,

wik =

1 if team i plays away in rounds

k and k+ 1,

0 otherwise.

A total of more than 7,900 variables are included.

Objective FunctionTo conform with ANFP requirements, our objec-tive function maximized the concentration of gamesbetween teams in the same group toward the finalrounds of the tournament.

The experience of previous tournaments suggests itis not advisable to overload the last round because bythat time the playoff qualifying teams tend to alreadybe determined. More specifically, it was requestedthat the majority of games involving teams of thesame group are concentrated between the 14th and19th rounds, that not more than three of these gamesbe scheduled for the last round, and that, to the extentpossible, none of them be played before the 10thround.

With the foregoing in mind, a game between teamsof the same group played on the final round wasassigned a weight of 15, while all other games heldafter the 9th round were assigned a weight equal tothe round number. Thus, the objective function usedin the model was the following:

max{ ∑

10≤k≤18

∑e

∑i∈t�e�

∑j∈t�e�

k · xijk+∑e

∑i∈t�e�

∑j∈t�e�

15 · xij19}�

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where

t�e� denotes the set of teams in group e�

e ∈ E = �1�2�3�4�

In the 2005 opening tournament, the objective func-tion incorporated a weighted sum of the round num-bers for the games between teams of the same group.In the 2005 closing tournament, games between rele-gation rivals were also weighted (a set of six candi-dates was deduced by the performance of the teamsin the opening tournament).

ConstraintsThe formulation of the constraints is given below.They are numbered to match the numbering of theconstraints described in §3.

∑k

�xijk+ xjik�= 1 ∀ i� j ∈ I� (1)

∑j

�xijk+ xjik�= 1 ∀ i ∈ I� k ∈K� (2)

9≤∑j

∑k

xijk ≤ 10 ∀ i ∈ I� (3)

∑k<19

yik ≤ 1 ∀ i ∈ I (4)

To this must be added the constraint relating they variables to the x variables:

∑j

�xijk+ xij�k+1��≤ 1+ yik ∀ i ∈ I� k < 19�

∑k<19

wik ≤ 1 ∀ i ∈ I (5)

To this must be added the constraint relating the wvariables to the x variables:∑

j

�xjik+ xji�k+1��≤ 1+wik ∀ i ∈ I� k < 19�

∑j

�xijk+ xij�k+1��= 1 ∀ i ∈ I� k ∈A�(6)

∑k

∑j∈t�e�

xijk = 2 ∀ e ∈ E� i ∈ t�e�� (7)

∑i∈South

�xijk+ xij�k+1��≤ 1−wjk ∀ j ∈North� k < 19�

∑i∈North

�xijk+ xij�k+1��≤ 1−wjk ∀ j ∈ South� k < 19�(8)

wik ≤∑

j∈North

�xjik+ xji�k+1�� ∀ i ∈North� k < 19�

wik ≤∑

j∈South�xjik+ xji�k+1�� ∀ i ∈ South� k < 19�

(9)

wik ≤∑

j∈Center

�xjik+ xji�k+1�� ∀ i ∈Center� k < 19� (10)

∑k

�xhik+ xhjk�= 1

∀h �= i�h �= j� i=COLO� j =UCH� (11)∑i� j∈PopularTeams

∑�10>k ∨k>16�

xijk = 0� (12)

∑k

�xhik+ xjik�=∑k

�xhjk+ xijk��

h=CATO� i=COLO� j =UCH� (13)∑i∈North

xijk ≤ 1− ∑i∈South

xihk

∀ j� h ∈ PopularTeams� j �= h� k ∈K� (14)∑k≤5

∑j∈OutlyingTeams

xjik = 0 ∀ i ∈ PopularTeams� (15)

∑j∈StrongTeams

�xijk+ xjik+ xij�k+1�+ xji�k+1��≤ 1

∀ i ∈ I� k < 19� (16)∑j∈PopularTeams

∑�6>k∨k>18�

�xijk+ xjik�= 0� i=CBLOA� (17)

∑h

�xihk+ xjhk�= 1 ∀ �i� j� ∈CrossedTeams� k ∈K� (18)

∑�8>k∨k>18�

�xijk+ xjik�= 0 ∀ �i� j� ∈RegionalClassics� (19)

2≤ ∑i∈Santiago

∑j

xijk ≤ 4 ∀k ∈K� (20)

∑6>k

xijk+ xjik = 0 ∀ i� j ∈D� (21)

∑j∈PopularTeams

∑k≤5

xijk ≥ 1 ∀ i ∈ TouristTeams� (22)

∑i

xijk = 1 ∀ j ∈ S� k= 1� (23)

∑e

∑i∈t�e�

∑j∈t�e�

xijk ≤ 3� k= 19� (24)

∑k

∑j∈North

xijk ≥ 1 ∀ i ∈North�

∑k

∑j∈South

xijk ≥ 1 ∀ i ∈ South (25)

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In total, the model considers around 3�000 con-straints.

Now let us examine certain additional aspects offormulating the constraints on patterns. We shall callP a set of 20 patterns and Home(k) the subset of theset of patterns that assigns home games in round k.Consider the following variables:

zip =

1 if pattern p is assigned to team i,

0 otherwise.

The constraints on the patterns are as follows:Exactly one pattern is assigned to each team:

∑p

zip = 1 ∀ i ∈ I (26)

Exactly one team is assigned to each pattern. Thisconstraint is useful to strengthen the relaxation of theinteger problem and holds only if the number of pat-terns in P is equal to 20. Alternatively, one mightconsider more patterns in this set. This might lead tobetter solutions but at the expense of higher runningtimes: ∑

i

zip = 1 ∀p ∈ P (27)

A team plays at home in a given round if theassigned pattern so indicates; otherwise, it plays away(Associated Press 2004):

∑j

xijk =∑

p∈Home�k�

zip ∀ i ∈ I� k ∈K (28)

Given the complexity of the solution, we assumethat it may be useful to incorporate cuts that helpreduce the size of the feasible domain of the relaxedproblem.

Note that because each group contains five teams,there can be at most two games between teams of thesame group for each group and round. We thereforeadd the following cut:

∑i∈t�e�

∑j∈t�e�

xjik ≤ 2 ∀ e ∈ E�k≥ 10 (29)

Finally, we add the following upper bound for theobjective function. Note that this condition is impliedby constraints (1), (24), and (29), but its inclusionshowed to be efficient in our experiments.

Given that the total number of games betweenteams of the same group is 40, the value attainable bythe objective function is upper bounded by 643:

∑10≤k≤18

∑e

∑i∈t�e�

∑j∈t�e�

k · xijk+∑e

∑i∈t�e�

∑j∈t�e�

15 · xij19

≤ 3 · 15+ 8 · 18+ 8 · 17+ 8 · 16+ 8 · 15+ 5 · 14= 643 (30)

AcknowledgmentsWe thank the anonymous referees for the valuable sug-gestions that improved this work. We also thank theAsociación Nacional de Fútbol Profesional de Chile, andin particular Alejandro Carmash and Felipe Chaigneau,the ANFP’s operational manager and executive secretary,respectively, during the period this project was carried out.This project would not have been possible without theirhelp. Finally, we are grateful to the Complex Engineer-ing Systems unit of the Millennium Sciences Institute forfinancial support in the completion of this work. FONDE-CYT grant number 1050747, Conicyt, Chile, and UBACyTX184, Universidad de Buenos Aires, Argentina partiallyfinanced the first author. The first and sixth authors receivedpartial funding from PROSUL 490333/2004—4, CNPq,Brazil.

ReferencesAnagnostopoulos, A., L. Michel, P. Van Hentenryck, Y. Vergados.

2006. A simulated annealing approach to the traveling tourna-ment problem. J. Scheduling 9 177–193.

Associated Press. 2004. Husband-wife team out bid after 24years. Retrieved July 23, 2007 http://sports.espn.go.com/mlb/news/story?id=1936328&CMP=OTC-DT9705204233.

Bartsch, T., A. Drexl, S. Kröger. 2006. Scheduling the professionalsoccer leagues of Austria and Germany. Comput. Oper. Res.33(7) 1907–1937.

Bean, J., J. Birge. 1980. Reducing traveling costs and player fatiguein the National Basketball Association. Data Mining KnowledgeDiscovery 10 98–102.

Cardemil, A., G. Durán. 2004. Un algoritmo tabú search para eltraveling tournament problem (in Spanish). Revista Ingenieríade Sistemas 18(1) 95–115.

Costa, D. 1995. An evolutionary tabu search algorithm and theNHL scheduling problem. INFOR 33 161–178.

Della Croce, F., D. Oliveri. 2006. Scheduling the Italian foot-ball league: An ILP-based approach. Comput. Oper. Res. 33(7)1963–1974.

Easton, K., G. Nemhauser, M. Trick. 2001. The traveling tourna-ment problem: Description and benchmarks. Proc. 7th Internat.Conf. Principles and Practice of Constraint Programming. Paphos,Cyprus, 580–584.

Easton, K., G. Nemhauser, M. Trick. 2003. Solving the travel-ling tournament problem: A combined integer programming

INFORMS

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Add

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htsan

dpe

rmission

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isav

ailableat

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s.org/.


and constraint programming approach. E. Burke, P. De Caus-maecker, eds. Lecture Notes in Computer Science, Vol. 2740.Springer, Berlin/Heidelberg, Germany, 100–109.

Henz, M. 2001. Scheduling a major college basketball conference—Revisited. Oper. Res. 49 163–168.

Nemhauser, G., M. Trick. 1998. Scheduling a major college basket-ball conference. Oper. Res. 46(1) 1–8.

Paenza, A. 2006. Matemática Estas Ahí? Episodio 2 (in Spanish).Siglo XXI, Buenos Aires, Argentina, 194–206.

Ribeiro, C. C., S. Urrutia. 2006. Heuristics for the mirrored travelingtournament problem. Eur. J. Oper. Res. 179(3) 775–787.

Schreuder, J. 1992. Combinatorial aspects of construction of compe-tition Dutch professional football leagues. Discrete Appl. Math.35 301–312.

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Scheduling the Chilean Soccer League by Integer Programmingmate.dm.uba.ar/~gduran/docs/papers/OR_Chilean_soccer.pdf · 2008. 6. 7. · Duránetal.:Scheduling the Chilean Soccer League