Scheduling Argentina’s professional basketball leagues: A variation on the Travelling Tournament Problem

European Journal of Operational Research 275 (2019) 1126–1138

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Innovative Applications of O.R.

Scheduling Argentina’s professional basketball leagues: A variation on the Travelling Tournament Problem Guillermo Durán a,b,c,∗, Santiago Durán b,c,d, Javier Marenco e, Federico Mascialino b, Pablo A. Rey f a

Departamento de Ingeniería Industrial, Universidad de Chile, Chile Departamento de Matemática and Instituto de Cálculo, FCEN, UBA, Argentina CONICET, Argentina d Université Paul Sabatier, Toulouse, France e Instituto de Ciencias, Universidad Nacional de General Sarmiento, Argentina f Departamento de Industria and Programa Institucional de Fomento a la Investigacion, Desarrollo e Innovación, Universidad Tecnológica Metropolitana, Santiago, Chile b c

a r t i c l e

i n f o

Article history: Received 2 November 2017 Accepted 13 December 2018 Available online 18 December 2018 Keywords: OR in sports Scheduling Integer programming Travelling tournament Problem

a b s t r a c t Operations research methods are applied to design the season schedules of Argentina’s professional basketball leagues using a format adopted in 2014 by the top two divisions. Following the setup used by the National Basketball Association (NBA) in North America, games are played any day of the week and away games are scheduled in road trips of one to four consecutive games. The main scheduling objective is to reduce the teams’ total travel distance compared to previous season formats through the use of predetermined trips submitted by the teams. The mathematical form of the problem is a variation on the well-known Travelling Tournament Problem. The modelling is divided into two successive stages, the first one defining the sequences in which each team plays the other teams and the second one assigning the days on which each game is played. Both stages use integer programming models that incorporate a series of constraints reflecting criteria requested by the Argentine Basketball Club Association. Implementation of the models has generated average travel distance reductions of more than 30% per away game, with consequential benefits in lower travel costs and less player fatigue. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Basketball is one of the most popular sports in Argentina today. The creation in 1985 of the National League (LNB by its Spanish initials) was instrumental in boosting interest in the sport throughout the nation. The founding of the LNB and the rise in the late nineties of a generation of outstanding players led by Emanuel Manu Ginobili (the recently retired star of the NBA’s San Antonio Spurs) have made Argentina one of basketballs world powers. The national team’s gold medal performance at the 2004 Athens Olympics on top of their second-place finish at the 2002 FIBA World Championship in Indianapolis, followed by the bronze medal at the Beijing Olympics in 2008, brought the country to the heights of international basketball glory. ∗ Corresponding author at: Departamento de Matemática and Instituto de Cálculo, FCEN, UBA, Argentina. E-mail addresses: [email protected] (G. Durán), [email protected] (S. Durán), [email protected] (J. Marenco), [email protected] (F. Mascialino), [email protected] (P.A. Rey).

https://doi.org/10.1016/j.ejor.2018.12.018 0377-2217/© 2018 Elsevier B.V. All rights reserved.

The LNB, the First Division of Argentine basketball, is a professional league that has expanded across the length and breadth of the country’s extensive territory. Unlike soccer, the nation’s number one sport whose professional activity is concentrated in the major cities (Greater Buenos Aires, Rosario, Santa Fe, Cordoba), basketball has also developed vigorously in smaller, regional centres. For example, in the 2016–2017 season, the 20 teams in the LNB were spread across 10 different provinces. This may be compared to the 30 First Division soccer clubs in the same season, which represented only 8 provinces and 26 of which were concentrated in only four urban areas. The country’s principal basketball teams have large budgets and in many cases receive financial support from provincial and municipal governments, major companies or professional soccer clubs. The sport also enjoys good media coverage. For example, three LNB games and one game of the Torneo Nacional de Ascenso (TNA), as the Second Division is known, are televised every week. Until 2013–2014 both the LNB and the TNA used manual methods to define their schedules. This often resulted in game calendars that required the teams to travel long distances over the course of

G. Durán, S. Durán and J. Marenco et al. / European Journal of Operational Research 275 (2019) 1126–1138

the regular season, with the consequential negative effects on the players. The need for modern techniques of sports scheduling was becoming ever more evident. In the present work, we apply mathematical programming methods to design the schedules for a season format adopted in 2014 by Argentina’s professional basketball leagues at the suggestion of the authors in the hope of reducing the team’s travel distances. Under this new setup, games are played any day of the week and away matches are scheduled in road trips. The mathematical form of the problem is a variation on the well-known Travelling Tournament Problem, defined in Easton, Nemhauser, and Trick (2001). The remainder of this article is organized as follows. Section 2 presents a brief review of the literature on sports scheduling. Section 3 describes the former and current season schedule formats for the two professional basketball divisions in Argentina. Section 4 defines the problem to be solved. Section 5 introduces the integer programming models used to solve the problem for both divisions and sets out the computational results. Section 6 analyzes the impact of the savings in team travel achieved by the new format. Section 7 discusses the results, presents some conclusions and shows our current work. Some variations on the models and additional computational results are given in the online appendix. 2. Literature review A wide variety of scheduling solutions to real-world cases have been reported in the literature over the last 25 years for a range of sports, including soccer (see Durán et al., 2007; Durán, Guajardo, & Sauré, 2017; Goossens & Spieksma, 2009; Rasmussen, 2008; Ribeiro & Urrutia, 2012), and basketball (see Nemhauser & Trick, 1998; Westphal, 2014; Wright, 2006). Most sports leagues around the world use a round-robin schedule format, in which each team plays every other team a set number of times. According to Nemhauser and Trick (1998) there are two types of round-robin schedules: temporally constrained and temporally relaxed. In temporally constrained schedules the number of available game slots (known as “rounds”) is equal to the number of games each team must play, plus any necessary byes for leagues with an odd number of teams. We therefore say that the schedule is compact. Many leagues use this setup, including most professional soccer leagues in Europe and South America. In temporally relaxed schedules, on the other hand, the number of available rounds for each team is greater than the minimum necessary so each team will have multiple byes. This type of schedule is used by professional leagues in North America, such as the National Basketball Association (NBA) and the National Hockey League (NHL) (see Costa, 1995; Craig, While, & Barone, 2009; Fleurent & Ferland, 1993); several amateur leagues (see e.g. Knust, 2010), and cricket leagues in Australia, England and New Zealand (see Willis & Terrill, 1994; Wright, 1994; Wright, 2005). There is little in the literature on the scheduling of the world’s most popular professional basketball league, the NBA. One study going back many years contains suggestions for improvements to the schedules as they existed at the time with a view to reducing travel costs (Bean & Birge, 1980). This was to be accomplished by implementing a heuristic that first generates possible road trips for all of the teams and then combines them into a feasible schedule. A more recent work proposes some ideas on using integer and constraint programming to obtain a method for generating NBA schedules automatically (Bao, 2009) and mentions a consulting firm responsible for designing the schedules that uses what is essentially a manual mechanism. An excellent test bed for different models, algorithms and methodological tools in sports scheduling is the Travelling

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Tournament Problem (TTP) (Easton et al., 2001). Given a set of n teams and the distances between their home cities, the TTP consists in scheduling a fictitious compact double round-robin tournament (each team plays every other team twice, once at home and once away) with 2(n − 1 ) rounds in such a way that no team plays fewer than L or more than U consecutive games either at home or away, no team plays any other team in two consecutive rounds and the total distance travelled is minimized. Also, no team returns home during a sequence of away games, known as a road trip. The TTP is very difficult to solve and its computational complexity has not yet been fully determined. What is known is that the problem is polynomial for L = 1 and U = 2 (Easton, Nemhauser, & Trick, 2003), and NP-hard for L = 1 and U = 3 (Thielen & Westphal, 2011) as well as for L = 1 and U = ∞ (Bhattacharyya, 2016), but for other values of L and U it remains open. Distance minimization and breaks maximization are equivalent if all distances are equal to 1 (Urrutia & Ribeiro, 2006). The first application of the TTP to a real case was reported in Bonomo, Cardemil, Durán, Marenco, and Sabán (2012) for Argentine volleyball. In Bao (2009), the author develops a temporally relaxed variation of the TTP called the TRTTP (Time-Relaxed Travelling Tournament Problem). In this problem, L and U are defined as in the TTP. The number of byes per team in the schedule is controlled by a parameter K. When K = 0, the problem just reduces to the TTP. In general, the computational complexity of the TRTTP is unknown. For the case where U = 1 the problem is trivial, and the author presents an algorithm that proves it is polynomial for any K with L = 1 and U = 2. For other values of L, U and K it is conjectured that the problem is NP-hard. Various instances generated for the TTP have also been studied for the TRTTP (Brandao & Pedroso, 2014). 3. Recent history of professional basketball tournaments in Argentina The Argentine Basketball Club Association (AdC by its Spanish initials) is the body that organizes the tournaments of the First and Second Division. In 2013–2014 the First Division had 16 teams and a regular season consisting of two phases, one regional and the other national. In the regional phase, the teams were divided into two 8-team zones (Northern and Southern) within which each team met every other one twice, once at home and once away. The national phase that followed used the same format but for the entire 16 teams as a single group. In all, each team played 44 regular season matches. The games were all scheduled on Fridays and Sundays using a team-pair system in which the teams were divided into groups of two (for a similar format used in the past see Campbell and Chen (1976). Each weekend, a pair A = (A1 , A2 ) would visit a pair B = (B1 , B2 ). On the Friday, A1 faced B2 while A2 took on B1 , and then on the Sunday the combinations were reversed. One weekend in the regional phase and a second one in the national phase would be devoted exclusively to intra-pair games, that is, matchups in which the member teams of each pair played each other twice. To ensure the system was workable the two teams in each pair were geographically close together, thus the visiting pair would have relatively short distances to travel between the Friday and the Sunday matches. At the end of the regular season, the 12 top teams in the standings fought for the league title in a best-of-five playoff system. No teams were relegated to the Second Division at the end of the 2013–2014 season. In the Second Division in 2013–2014 there were 20 teams. As with the First Division the season was divided into two parts, but in this case they consisted of zonal and regional phases. In the zonal phase, there were four groups of 5 teams, each team

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playing every other one within its group at home and away with one game played each weekend. In the regional phase there were two groups (Northern and Southern) of 10 teams, each team playing every other one at home and away, again with one game each weekend. In all, each team played 26 games during the regular season. At the end of the regular season, the 8 top teams in each group battled for two promotions to the First Division in a bestof-five playoff system. The last (tenth) place team in each group was relegated to the Torneo Federal, Argentine basketball’s Third Division. Neither the First Division’s team-pair format nor the Second Division’s weekly game setup was very difficult to schedule using manual methods, but a serious drawback in both arrangements was that they took little advantage of team journeys to other venues to play multiple consecutive games on a single trip. As a result, the teams travelled more kilometres over the course of a season than was necessary, with negative consequences in terms of travel costs and player fatigue. For this reason the AdC decided, on the suggestion of the present authors, to switch to a new scheduling format for the 2014–2015 season in both divisions, similar to that of the NBA. Unlike the European leagues, the conditions facing the NBA schedulers more closely resemble those of Argentina in that in both cases, the distances between the teams’ home cities are great and the number of games in a season is relatively high. Applying the NBA format to the Argentine leagues would mean, for example, that the team from the city of Formosa could travel to Buenos Aires 1200 kilometers to the south and play 3 consecutive away games against teams based there before returning home. In addition to generating very considerable travel cost savings, this format would mean abandoning the team-pair system and weekend-only scheduling. The teams themselves would submit their own road trip preferences that did not necessarily have to minimize total distance but could couple some travel savings for the club with other benefits based on sporting criteria they might also find desirable. In 2014–2015 the First Division grew from 16 teams to 18. Each team played 18 games in the regional phase and 34 games in the national phase. Once again, there were no relegations to the Second Division. For the 2015–2016 season the number of First Division teams again grew by 2, for a new total of 20. Each side played 56 games, 18 regional games followed by 38 national games. The First Division teams in 2015–2016 and the relative locations of their home cities are shown on a map of Argentina in Fig. 1. A road trip format was also adopted by the Second Division in 2014–2015, when there were 24 teams. The teams each played a total of 34 games over the entire regular season. The two promotions to the First Division were decided by various best-of-five playoff series at the end of the season. No teams were relegated to the Third Division. In 2015–2016 the number of teams in the Second Division was 26 and the format was similar to the previous year. Each team played 36 games during the regular season. Only one side was promoted to the First Division, after it won several best-of-five playoff series. Once again, no teams were relegated to the Third Division. The Second Division teams in 2015–2016 are also shown on a map of Argentina in the online appendix. 4. Description of the problem For both the First and Second divisions, the problem to be solved consists in scheduling the tournaments so as to assign each team the road trip requested by the teams and the AdC. At the beginning of the planning period, each team informs a set of preferred trips. Teams are encouraged to submit as many redundant trips as possible since such redundancy improves the chances of finding solutions. When we consider that it is necessary we ask

the AdC to add “reasonable” trips for each team to help ensure the feasibility of the model. Teams also inform about their preferred weekdays for playing home. Since a certain number of byes are allowed, the schedules will be temporally relaxed. In this sense our problem resembles the TRTTP formulation, but since the road trips are defined in advance by the teams themselves, what we have is a variation on the TRTTP. We denote this new variant the Trip Preferences-Time Relaxed Travelling Tournament Problem (TP-TRTTP). The TRTTP is a special case of TP-TRTTP since an instance of TRTTP is also an instance of the TP-TRTTP in which any road trip is allowed. Note that trip preferences can also be incorporated into a TTP. Road trips in our problem usually consist of 2 or 3 consecutive away matches before returning home. Occasionally there may also be a single-game trip or a four-game trip. Other constraints are applied to prevent long consecutive sequences of home or away games, avoid home games when a team’s stadium is not available, take into account the requirements of the television broadcaster, and schedule attractive games at crucial moments during the season. With the implementation of the new formats in the two top divisions, it was decided to include rest weeks at certain points during the season. In the First Division, one such week is inserted into the regional phase and two more in the national phase while in the Second Division a single rest week separates the two phases. These weeks are used a posteriori for any reschedulings that may be required if games must be suspended to allow teams to participate in international matches whose dates are not known when the season schedule is finalized, or for any other unforeseen circumstance. Since these modifications tend to be relatively simple, they are handled manually. As for the distribution of the two divisions’ teams into zones and conferences, this was done for the most part by the AdC following natural geographic criteria without resorting to technical methods. In a few cases, however, where the solution was not obvious, we made recommendations based on heuristic methods that reduced the teams’ global travel distances. Given the computational difficulties of solving in a single model both the order of the games and the specific days they are played, it was decided to tackle the problem in two stages. Thus, the first stage assigns each team’s game sequence with approximate playing days and the second stage determines the day of each game, taking into account the teams’ expressed preferences for home-game days. The models for each stage are introduced in the next section.

5. Solution approach and computational results In this section, we set out our solution approach, which is built around a separate integer programming model for each of the two stages just mentioned. The model for the first stage assigns the order of each team’s games, considering that the away games are to be scheduled in sequences known as road trips (hereafter simply “trips”). The second stage uses a model to determine the exact day assigned to each game while maintaining the game order decided in the first stage. Since the various phases, or zones within phases, that make up the seasons of the two basketball divisions as discussed above are completely independent of one another, they will also be scheduled independently and will be referred to collectively as “season units”. The two models were implemented in the Zimpl programming language and solved by CPLEX 12.6.3 on a computer with an TM Intel® Core i7-5930K processor and 16 gigabytes of RAM running Ubuntu 16.04 LTS. The main computational results of the models are also presented in this section.

G. Durán, S. Durán and J. Marenco et al. / European Journal of Operational Research 275 (2019) 1126–1138

Formosa La Unión

Santiago del Estero Olímpico Quimsa

Corrientes San Martín Regatas

Sunchales

Concordia

Libertad

Córdoba Instituto Atenas

Estudiantes

Paraná Sionista Ciudad Autónoma de Buenos Aires

Junín Argentino

Obras Sanitarias Boca Juniors Lanús San Lorenzo de Almagro Ferrocarril Oeste

Mar del Plata Bahía Blanca

Quilmes Peñarol

Bahía Basket

Comodoro Rivadavia Gimnasia Indalo

Fig. 1. First Division (LNB) teams, 2015–2016 season.

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5.1. First-stage model: determination of trips and sequencing of games The first-stage model takes as its main input the possible trips for each team. It does not define the actual days on which the games will be played but does consider that up to three games per week may be scheduled, which for purposes of this stage are limited to Mondays, Wednesdays and Fridays (in the second stage, the model will assign games on any day of the week). In any given week, however, a team will not necessarily play in all three rounds since the total number of rounds at the rate of three per week would be about 50% more than the total number of games each team will play in the season. The idea is in fact to schedule in such a way that the teams each play an average of twice a week, meaning they will also have an average of one weekly bye. With these characteristics in mind, the objective of this first model is to maximize the number of games that are assigned to rounds coinciding with the teams’ trip preferences. The data inputted to the model and the parameters representing them are as follows: • •

•

•

•

The set E of teams. For each team i ∈ E, a set T(i) of possible trips. Each trip t is a sequence of 1 to 4 different teams other than i. The number of games in t is |t| and the set of all trips is T = ∪i∈E T (i ). Also, for each team i, T (i ) = P T (i ) ∪ RT (i ) where PT(i) is the set of preferred trips the team has chosen, and RT(i) is the set of “reasonable” trips suggested by the AdC. For example, one team may choose the trip A-B-C (with exactly this order), but may not choose the trip C-B-A due to particular preferences (as, e.g., logistic or competitive considerations). In these cases, the trip C-B-A can be added as “reasonable”, which may not be an ideal trip for the team but nevertheless is equivalent to the original one in terms of traveled distance. These trips (included in T(i)) with their corresponding strict order of games constitute a covering of all of its away games. The games making up a trip are assumed to be played in consecutive rounds. Observe that trips are not necessarily maximal sequences of visits and that it is completely legal to include in set T(i) trips that are subsequences of other trips in T(i). The total number n of rounds. The set of all rounds is denoted F = {1, . . . , n}. The set of games to be played at home and away by each team i ∈ E in the season unit to be scheduled. The total number of games to be played by each team is strictly less than n, implying that each team has a set of byes over the course of the season unit. The set C of cutoff dates. This set, predefined by the AdC, reflects the existence of the rest weeks inserted into the season calendar. A trip must therefore not start on a date that would not allow enough time to complete it before the start of a rest week. The cutoff date for each rest week is expressed in the model as the Friday immediately preceding it.

The model contains a binary variable ztk for each trip t ∈ T and round k ∈ F where k ≤ n − |t | + 1 such that ztk = 1 if and only if trip t starts with round k. In addition, we introduce the auxiliary variable xijk for each pair of teams i, j ∈ E, i = j and each round k ∈ F such that xi jk = 1 if and only if team i plays at home against team j in round k. Although the x variables are defined in terms of z and are therefore not strictly required, their presence allows us to simplify the formulation of the model. With the above elements we can now formally state the firststage model. The objective function seeks to maximize the number of games that are scheduled in accordance with the teams’ trip

preferences:

max

  

|t | ztk .

i∈E t∈PT (i ) k∈F

The constraints are the following. 1. All games must be scheduled.



∀ i, j ∈ E, i = j.

xi jk = 1

(1)

k∈F

For the four-team zonal groups of the TNA 2014–2015, which are quadruple round-robin tournaments, the right-hand side of these constraints was set to 2. 2. Each team plays a maximum of one game per round. Note that this constraint implies that no team is involved in more than one trip at the same time.





xi jk + x jik ≤ 1

∀ i ∈ E, ∀ k ∈ F.

(2)

j∈E

3. The games played are chosen from the set of possible trips.



xi jk =

∀ i, j ∈ E, i = j, ∀ k ∈ F.

zt,(k−pos(t,i ))

(3)

t∈T ( j )

In view of the definition of xijk stated above, the value of the variable will be 1 if and only if team j is assigned a trip that includes a game against i in round k. Given a team i ∈ E and a trip t ∈ T that includes team i, we define pos(t, i ) ∈ {0, . . . , |t | − 1} as the position of i in the games sequence of trip t. 4. Each team plays at least one game at home in the four rounds following each trip. 3 

∀ i ∈ E, ∀ t ∈ T (i ), ∀k ∈ F , k + |t | + 3 ≤ n.

xi j,k+|t |+s ≥ ztk

j∈E s=0

(4)

This constraint ensures teams are not assigned excessively long consecutive trips. This constraint is weaker than asking no two consecutive trips in the schedule, and was included instead of the latter (stronger constraint) in order to improve the chances of solving the resulting model. 5. No trip can start before and end after a cutoff date.

ztk = 0

∀ c ∈ C, ∀ t ∈ T , ∀ k ∈ F , k ≤ c < k + |t | − 1.

(5)

6. No trip can be assigned for which there are not enough rounds left in the season unit for it to be completed.

ztk = 0

∀ t ∈ T , ∀ k ∈ F , k > n − |t | + 1.

(6)

7. Each team has a bye before or after each trip to allow the players to rest, or else the trip ends in the last round of the season unit.



(xi j,k+|t | + x ji,k+|t | ) +

j∈E



(xi j,k−1 + x ji,k−1 ) ≤ 2 − ztk

j∈E

∀ i ∈ E, ∀ t ∈ T (i ), ∀ k ∈ F , 1 < k ≤ n − |t |.

(7)

8. No team can have more than three byes in a row, to avoid extended periods without a game. 2 

xi j,k+s + x ji,k+s ≥ 1

∀ i ∈ E, ∀ k ∈ F , k + 2 ≤ n. (8)

j∈E s=0

9. A team must have at least one away game every  + 1 rounds.  

x ji,k+s ≥ 1

∀ i ∈ E, ∀ k ∈ F , k +  − 1 ≤ n.

(9)

j∈E s=0

This constraint prevents the “saturation” of local fans. Usually we consider l = 5.

G. Durán, S. Durán and J. Marenco et al. / European Journal of Operational Research 275 (2019) 1126–1138

10. Each team plays at least one game in the first three rounds. 3 

∀ i ∈ E.

(xi jk + x jik ) ≥ 1

(10)

j∈E k=1

11. In the last round all teams must play, except that if the number of teams is odd, all teams but one must play. This is a scheduling fairness condition imposed by the AdC.



xi jn ≥

|E | − 1 2

i∈E j∈E

.

(11)

12. Nature of the variables:

ztk ∈ {0, 1}

∀ t ∈ T, ∀ k ∈ F, ∀ i, j ∈ E, i = j, ∀ k ∈ F.

xi jk ∈ {0, 1}

The structure of the problem just described makes it difficult to solve double round-robin cases with more than 13 teams. This is consistent with what is known about both TTP and TRTTP, for which there are certain instances where n = 12 and n = 10, respectively, that are still open problems (see Trick, 2016). To get around this limitation for larger instances, the corresponding season unit is partitioned using a procedure described in the next subsection. 5.1.1. Partition of long season units Rest weeks are used as dividers to break the schedules into separate (though not independent) blocks. This means that once a game is scheduled in a given block it cannot be scheduled in any other block. Let C = {c1 , . . . , c p } be the set of cutoff dates and define c0 = 0 and c p+1 = n. The above considerations suggest the following heuristic algorithm, which has been used for the largest instances and in every case has generated satisfactory solutions: Step 1. Divide the set of rounds F into subsets F0 , . . . , Fp , defined as Fq = {cq + 1, cq + 2, . . . , cq+1 }, for q = 0, . . . , p. Thus, the rounds in each Fq are consecutive and cover the interval between successive cutoff dates, constituting the (q + 1 )th block of the season unit. Step 2. Define P := ∅. This set represents games that have already been played in previous blocks. Step 3. Define an ordering to solve the various blocks making up the season unit. This ordering is represented by the permutation σ of the set {0, . . . , p}. For q ∈ {0, . . . , p}, execute the following iteratively: (a) Solve the Section 5.1 model but with F = Fσ (q ) , considering constraint (11) only for Fp and replacing constraint (1) by



xi jk ≤ 1

∀ i, j ∈ E, i = j

(meaning that now not all games will be scheduled), eliminating trips that include games from set P that have already been played and replacing the objective function with the following expression:

max

   i∈E

t∈PT (i ) k∈F

2|t |ztk +

 

Step 4. When the algorithm has completed all of the blocks in a season unit, combine the schedules generated for each one and return the complete solution thus obtained. The idea behind the above procedure is to design a schedule for a season unit by blocks, using the cutoff dates as the dividers. The model that solves each block (step 3a in the algorithm) attempts to maximize the number of games in the block, thus facilitating the task for the following blocks. As it was set up in the step 3 of the algorithm, the different blocks making up a season unit are not necessarily solved sequentially. In other words, a non-chronological order may be defined by permuting the blocks. The permutation for the chronological case is σ equal to the identity function. If the solution obtained by the algorithm for the complete set of blocks leaves out some of the games, the procedure can be iterated by repeating the steps applied to the various blocks with additional constraints in order to generate different solutions for each block. For example, the assignment of a game to a given round in the original solution for a block could be disallowed, thus forcing a different assignment. In effect, this is a backtracking algorithm that would run until a solution containing all of the games is finally produced. For instances where it was difficult even to obtain feasible solutions whether or not partitioning was used, we implemented a model with penalties, developed in the online appendix. The national phase instances of the First Division for 2014–2015 and 2015–2016 were solved using the penalized model and partitioning into blocks. In these cases, we divided the units into thirds. In these seasons, there were two cutoff dates, then partition in thirds was a natural option. The order that gives the best results is to start by solving the last third (with the constraint that all the teams play in the last round), then solve the first third (with the constraint limiting the number of byes in the early rounds), and finally, solve the second third, which has no strong boundary condition requested by the AdC. The maximization of the numbers of games in the first solved blocks may result in an unbalanced schedule in terms of the final number of games in each block. To avoid this, we add constraints that limit the numbers of home, away and total games each team plays in each block. To formulate the constraint for home games we first define two parameters, Lq and Lq , which indicate respectively the minimum and maximum number of home games to be played up to and including block q. The inequalities added for team i are:

Lq − JLiq ≤

 |t |ztk .

t∈RT (i ) k∈F

In other words, what is to be maximized is the number of games chosen for the interval of rounds (i.e., block), double-counting games that figure in the teams’ preferred trips. Further constraints added to avoid blocks that are unbalanced in terms of the number of games to be played are discussed later in this subsection. (b) Add to set P the games played in the solution of the previous step.



xi jk ≤ Lq − JLiq ,

j∈E k∈F

or equivalently,

Lq ≤ JLiq +

k∈F

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xi jk ≤ Lq ,

j∈E k∈F

where JLiq is the number of home games team i plays in the blocks solved before block q. The constraints for away and total games are formulated analogously. The values of the Lq and Lq parameters are based on the average value per block. If, for instance, the complete season problem is solved in three blocks, each with approximately the same number of rounds, the values will be one-third of the total number of games and will depend on the order in which the blocks are solved. As an example, for the national phase of the First Division’s 2014–2015 season in which 18 teams each played 34 games, the values used are shown in Table 1. Recall that the last third (i.e., block) is solved first, followed by the first third and then the second one.

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Table 1 Upper and lower bounds on the number of accumulated games for defining the “regularity” constraints: National Phase, First Division, 2014–2015. Block

Home/Away

First solved block (last third) Second solved block (first third)

Total

L. Bound

U. Bound

L. Bound

U. Bound

5 10

9 14

11 20

14 24

Table 2 Results of first-stage model for First Division instances. Instance

Time (second)

Nodes

Total games

Pref. trip g.

Total dist. (kilometer)

LNB-2014–2015 1/3 LNB-2014–2015 2/3 LNB-2014–2015 3/3 Total LNB-2014–2015 LNB-2014–2015 North. LNB-2014–2015 North. LNB-2014–2015 South. LNB-2014–2015 South. LNB-2015–2016 1/3 LNB-2015–2016 2/3 LNB-2015–2016 3/3 Total LNB-2015–2016 LNB-2015–2016 North. LNB-2015–2016 North. LNB-2015–2016 South. LNB-2015–2016 South.

346 3630 2437 6413 46 60 288 149 6899 3061 15,043 25,003 512 575 264 1576

8 31,680 492

101 99 106 306 72 72 72 72 125 125 130 380 90 90 90 90

101 90 106 297 60 64 61 61 125 124 130 379 90 90 90 90

83,064 90,220 85,109 258,393 40,569 42,039 53,566 54,747 105,320 109,028 102,829 317,177 46,649 46,649 63,378 63,378

D P D P

D P D P

748 589 9143 2656 4318 22,413 16 9035 14,065 0 398

Also incorporated are the boundary conditions for the byes at the start and end of the blocks. For the First Division cases solved in three thirds with the second third solved last, we denote as B1 ⊂ E the set of teams that have a bye in the last round of the first third and as B3 ⊂ E the set of teams that have a bye in the first round of the last third. In the second third, the inequalities



(xi j1 + x ji1 ) ≥ 1 − b1i

i ∈ B1

j∈E

and



(xi j|F2 | + x ji|F2 | ) ≥ 1 − b3i

i ∈ B3 ,

j∈E

are added, where b1i and b3i are non-negative continuous vari ables with an upper bound of 1. Also, the terms − i∈B1 α b1i and  − i∈B3 α b1i are included in the objective function. The idea is to inhibit the scheduling for any team of a bye in both the last round of the first third and the first round of the second third, or in both the last round of the second third and the first round of the last third. The penalty coefficient α was set at 10. Analogously, penalized constraints were implemented to avoid creating extended home- or away-game sequences once the three blocks were joined together to obtain the solution for the complete scheduling unit. Finally, for cases solved in blocks, to reduce unnecessary scheduling of non-preferred trips and afford greater flexibility in the final block, only for this block were such trips considered. In the First Division cases solved in three blocks with the second one solved last, this generated schedules that avoided non-preferred trips at the start and the end of the season units. 5.1.2. Computational results This subsection presents all the results obtained with the model described above for the 2014–2015 and 2015–2016 seasons with the exception of the zonal phase of the Second Division’s 2014– 2015 season, which consisted of six groups of four teams. The schedules for these latter cases were easily generated using a slight variation on the model. For illustrative purposes the results for one of these groups are also reported. Solutions were found by the model without resorting to partitions for all instances save those of the First Division’s national

phase for both seasons. The values shown for these two cases are those obtained by step 3a of the partitioning algorithm set forth above. The stopping criterion was a gap of less than 0.1% for all but the partitioned units, where the gap applied was 10% in every case except the last solved stage for which it was reduced to 1%, a small figure being necessary here in order to achieve feasibility for the complete problem. For the instances that were handled as complete problems (i.e., without partitioning), a second model was also solved. In this alternative formulation the objective function was the usual one for the TTP, that is the minimization of the total distance travelled by the teams. To represent this each ztk variable was multiplied by the total distance of trip t (non-preferred trips were penalized by doubling their distances). The main characteristics of the instances solved by the firststage models are summarized in the appendix. The results for these instances are summarized in Tables 2 and 3. For the regional phase cases (Northern and Southern conferences), the letters “P” and “D” in the Instance column refer respectively to the solutions generated by the original form of the model, which maximized the number of games in preferred trips, and the alternative formulation, which minimized total distance travelled. The results for the two objective functions are in fact similar, which suggests that either one could be used without producing major variations in the final schedules. For the national phase cases, where the model maximized the number of games scheduled in preferred trips using partitions, the blocks are identified by their chronological order as “1/3”, “2/3” and “3/3”. The values shown for the blocks are those obtained when solving them in the order “3/3”,“1/3”,“2/3”, which gives the best outcomes. The first two data columns in the tables report the solution times in seconds and the number of branch-and-bound nodes used by CPLEX while the last three columns indicate the quality of the solutions in terms of the two objective functions. More specifically, the third and fourth data columns give the number of games scheduled and how many of them are preferred trip games. The final column displays the total distance travelled by all of the teams on the trips scheduled by the model. As can be seen, the model solution times depend on the total size of the season unit. However, the choice of objective

G. Durán, S. Durán and J. Marenco et al. / European Journal of Operational Research 275 (2019) 1126–1138

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Table 3 Results of first-stage model for Second Division instances. Instance TNA-2014–2015 TNA-2014–2015 TNA-2014–2015 TNA-2014–2015 TNA-2014–2015 TNA-2014–2015 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016 TNA-2015–2016

N.Reg. D N.Reg. P S.Reg. D S.Reg. P Group D Group P N.Reg. D N.Reg. P S.Reg. D S.Reg. P Centre D Centre P North. D North. P South. D South. P

Time (second)

Nodes

Total games

Pref. trip g.

Total dist. (kilometer)

806 1634 3153 10 0 0 <1 <1 23,842 12,156 14,265 3326 227 105 23 80 1 1

0 4805 8792 4155 0 0 31,511 4515 62,562 12,723 36,788 8165 2140 6255 0 0

132 132 132 132 24 24 156 156 156 156 72 72 42 42 42 42

132 132 132 132 24 24 152 156 156 156 72 72 42 42 42 42

78,973 79,084 170,159 170,159 11,616 11,616 105,233 105,625 168,943 169,044 35,610 35,950 30,150 30,150 47,112 47,112

function makes little difference either to solution times or to the two solution-quality indicators (number of games in preferred trips and total distance). In every instance, the models scheduled at least 90% of the games in trips the teams preferred. Finally, four additional runs were executed with the partition procedure on the national phases of the First Division to compare solution times and quality. These results are reported in the online appendix to this paper. 5.2. Second-stage model: assignment of game days Once a satisfactory solution defining the order of the games is obtained from the first-stage model described above, the games themselves are assigned to specific days by the second-stage model, presented in what follows. To the best of our knowledge, no mathematical model for assigning game days in a sports league season schedule after the order of the games has been fixed previously, has been reported in the sports scheduling literature. A phased approach is used in Kyngäs et al. (2017) for scheduling the Australian football league, but in a different way and for a different problem. The input to the second-stage model, in addition to the order of the games, consists of the calendar days designated for holding games and a list of each team’s preferences regarding the days on which it would prefer to play at home. More formally, the data inputted to the model and the parameters representing them are as follows: • •

•

•

The set E, defined in the previous subsection. The set D = {1, . . . , m} of designated calendar days for holding games, numbered continuously starting with a preset Day 1. The total number gi of games team i plays. Note that although this number is the same for all teams over an entire scheduling unit, if the latter is solved by partitioning then this parameter for a given block may vary from team to team. Sets Ji = {1, . . . , gi } of indexes for the order of the games to be played by each team i as determined by the first-stage model. Thus, for each team the index p represents its pth game.

The model is applied to complete season units in cases where the first-stage model solved the entire unit without partitioning, and to blocks where the earlier model did use partitions. The solution for a given block does not affect the solutions of the other blocks. The model contains a binary variable xipt for each team i ∈ E, each game p ∈ Ji and each date t ∈ D such that xipt = 1 if and only if team i plays its pth game on date t. It also contains the auxiliary binary variables daip and dbip , which are needed for cases

where the constraints reflecting the AdC’s requirement that there be two rest days before and after each trip cannot be satisfied. Thus, dbip = 1 if and only if team i has a single rest day before the first away game on a trip (the pth game). Analogously, daip = 1 if and only if the team i has a single rest day after the last away game on a trip (the pth game). In the objective function, a penalty is applied each time any of these variables equals 1. To express the teams’ day preferences, a parameter pref(i, t) is defined for each team i ∈ E and each date t ∈ D. It is equal to 1 if team i prefers playing at home on date t and 0 otherwise. Finally, a parameter home(i, p) is defined for each team i ∈ E and each game p ∈ Ji such that home(i, p) = 1 if and only if team i plays game p at home and 0 otherwise. With the above elements we can now formally state the second-stage model. The objective function seeks to maximize the number of games played on days a team prefers to be at home and penalizes the number of times teams start or end a trip without the number of requested rest days. To achieve the desired effect while avoiding numerical problems in computing the solutions, a value of 1,0 0 0 was set for the penalty coefficient.

 

max

pref(i, t )home(i, p)xipt − 10 0 0

i∈E,t∈D p∈Ji



(db pt + daip ).

i∈E,p∈Ji

The constraints are the following. 1. A day is assigned to each game.



xipt = 1

∀ i ∈ E, ∀ p ∈ Ji .

(12)

t∈D

2. The order of the games is not reversed and there are no more than r days between two consecutive games played by any given team (this condition also ensures that no team plays twice either on the same day or consecutive days). min(t+r,m )

xipt ≤



xi,p+1,s

∀i ∈ E, ∀ p ∈ Ji , p < gi , ∀t ∈ D, t ≤ m − 2.

s=t+2

(13) The value of r is set at 7, thus ensuring that no more than a week elapses between any two of a team’s games. 3. No team plays at home on a day its home stadium is not available (if this condition holds for a long period, a similar constraint should be also included in the first model for the corresponding set of rounds).

xipt = 0

∀ i ∈ E, ∀ p ∈ Ji , ∀ t ∈ D,

(14)

if team i plays a home game p and its stadium is not available on date t.

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4. Two teams whose home stadiums are in the same city do not play at home on the same day.

∀ i, j ∈ E, ∀ p ∈ Ji , q ∈ J j , ∀ t ∈ D,

xipt + x jqt ≤ 1

(15)

if two different teams i and j are based in the same city and play games p and q, respectively, at home. 5. A rest day must be scheduled between two consecutive away games on the same trip.

∀ i ∈ E, ∀ p ∈ Ji , p < gi ∀ t ∈ D, t < m − 1,

xipt = xi,p+1,t+2

(16)

where p and p + 1 are two consecutive games played by team i on the same trip. 6. We want to leave at least two rest days before and after each trip: (a) if p is the first game of a trip,

xi,p−1,t−2 + xipt ≤ 1 + dbip

∀ i ∈ E, ∀ p ∈ Ji , p > 1 ∀ t ∈ D, t > 2.

(17)

∀ i ∈ E, ∀ p ∈ Ji , p < gi ∀ t ∈ D, t < m − 1.

(18)

(b) if p is the last game of a trip,

xipt + xi,p+1,t+2 ≤ 1 + daip

7. Each game between two teams is assigned to the same day. This must be included to ensure consistency between the dates assigned to the same game given that the model determines them separately for each team.

∀ i, j ∈ E, ∀ t ∈ D

xipt = x jqt

(19)

if game p for team i and game q for team j are a game in which i and j play each other. 8. Televising of games. A set R ⊂ D of days is defined as the days on which a game is televised. The following constraint incorporates the television broadcaster’s requirement that at least one game be scheduled on each day in the set:



∀t ∈ R.

xipt ≥ 1

i∈E p∈Ji

9. In the last day all teams must play, except that if the number of teams is odd, all teams but one must play. This is a scheduling fairness condition imposed by the AdC and is related to constraint (11) in the first model.



xi,gi ,m ≥

|E | − 1

i∈E

2

.

(20)

10. Nature of the variables:

xipt ∈ {0, 1} daip ∈ {0, 1} dbip ∈ {0, 1}

∀ i ∈ E, ∀ p ∈ Ji , ∀ t ∈ D, ∀ i ∈ E, ∀ p ∈ Ji , ∀ i ∈ E, ∀ p ∈ Ji .

Unlike the case with the first-stage model, the structure of this problem does not prevent the solver from finding an optimal solution for every instance. Indeed, it has found a solution in a few seconds for the majority of instances, and no more than 11 minutes for the most complex ones. In only a few cases have the dait and dbit variables taken a value of 1, no more than 6 in any one instance, and when they do it means that there exists no feasible solution satisfying the rest day conditions before and after each trip. Details on the characteristics of the instances solved by this model are summarized in the online appendix.

6. Impact The two models developed in the previous section were used to design the schedules for the various phases of all the seasons since 2014–2015 for both divisions. Each instance was solved several times while varying certain constraints or parameters, in some cases using both the original and the alternative objective functions described above, and in certain instances using the first-stage penalized model. For each phase of the seasons, a set of solution proposals generated by the models was submitted for consideration by the AdC, which studied them for possible improvements and additional conditions. This decision process was iterative, continuing over a period of two or three weeks during which as many as 30 different proposals were made for each phase before the final version was arrived at. In the rest of this section we carry out a comparative analysis of the distances travelled by the different teams in the two divisions under the current road-trip based formats scheduled using our mathematical models and the previous team-pair formats scheduled by manual methods. Our first set of comparisons contrasts the First Division’s 2013–2014 season, the last one manually scheduled, with the mathematically scheduled 2014–2015 and 2015–2016 seasons. The relevant results are set out in Table 4. Using our models for the first time, the teams in 2014– 2015 travelled a total of 8% fewer kilometres than in 2013–2014 (353,365 kilometers versus 385,810 kilometers) even though in 2014–2015 they each played 4 more away games and there were two more teams in the Division. The teams’ average travel distance per away game in 2014–2015 was 31% lower (755 kilometers versus 1096 kilometers) than the year before. As for 2015–2016, the comparisons with 2013–2014 are broadly similar. Of the 15 teams that played in both of those seasons, there were 13 that travelled fewer kilometres in 2015–2016 even though in that year they each played six more away games. For just two teams, Argentino and Libertad, the reverse was true but the difference was very small, in particular for the latter. Furthermore, the teams’ average travel distance in 2015–2016 was almost 30% less than in 2013–2014 (772 kilometers versus 1096 kilometers). In Table 5, we compare the distances actually travelled by First Division teams in 2015–2016 as scheduled by our models with the distances they would have travelled that same season had the old team-pair format with manual methods been used. As the bottom row indicates, the model solutions dictated an average per-team travel distance for away games that was 22% lower (772 kilometers instead of 991 kilometers). As for the Second Division, we compare the distances actually travelled in 2014–2015 and 2015–2016 using our models with the distances that would have been travelled in those same years using the earlier manual methods when the teams played once a week and away games were not part of multi-game road trips. No reliable comparisons are possible for this Division between the two later seasons and 2013–2014, the last time the manual methods were used, because of the many changes over the three-year period in the Division’s mix of teams. The pertinent data for 2014–2015 are given in Table 6. As can be seen, the mathematically scheduled road trip format generated a total distance travelled that was almost 30% shorter than under the earlier format (329,030 kilometers versus 468,900 kilometers) while the average distance travelled per away game was more than 42% smaller (806 kilometers versus 1395 kilometers). The comparison is repeated in Table 7 for the 2015–2016 season. Once again, our mathematically scheduled format improved dramatically on the distances that would have been travelled under the old manual approach, greatly reducing the total number of kilometres while cutting the average per away game by more than 38% (840 kilometers versus 1360 kilometers).

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Table 4 Comparison of distances travelled by First Division teams in 2013–2014, 2014–2015 and 2015–2016. Team

2013–2014 season

2014–2015 season

2015–2016 season

Kilometers travelled

Away

Average

Kilometers travelled

Away

Average

Kilometers travelled

Away

Average

Argentino Ciclista Quilmes Peñarol Bahía Basket Gimnasia Indalo Boca Juniors Obras Sanitarias Lanús La Unión Regatas San Martín Quimsa Olímpico Atenas Libertad Sionista Estudiantes Instituto Ferro San Lorenzo

18,561 − 23,546 25,455 28,347 47,684 20,278 20,278 20,278 27,069 24,362 − 24,380 24,305 22,087 18,401 19,648 21,131 − − −

22 − 22 22 22 22 22 22 22 22 22 − 22 22 22 22 22 22 − − −

843.68 − 1070.27 1157.05 1288.50 2167.45 921.73 921.73 921.73 1230.41 1107.36 − 1108.18 1104.77 1003.95 836.41 893.09 960.50 − − −

15,255 16,826 19,335 19,485 17,426 37,218 15,632 17,360 18,814 22,013 20,121 20,484 19,367 19,649 22,008 18,127 15,419 18,826 − − −

26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 − − −

586.73 647.15 743.65 749.42 670.23 1,431.46 601.23 667.69 723.62 846.65 773.88 787.85 744.88 755.73 846.46 697.19 593.04 724.08 − − −

20,014 − 22,764 22,764 20,372 43,215 18,332 18,332 18,814 26,362 21,626 22,752 20,722 21,397 21,790 18,685 17,967 18,818 20,390 18,992 18,332

28 − 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28

714.79 − 813.00 813.00 727.57 1,543.39 654.71 654.71 671.93 941.50 772.36 812.57 740.07 764.18 778.21 667.32 641.68 672.07 728.21 678.29 654.71

Total

385,810

1096.05

353,365

755.05

432,440

772.21

Table 5 Comparison of distances travelled by First Division teams in 2015–2016 using road trip format (mathematical models) and team-pair format (manual methods). Team

2015–2016 season Team-pair format (Manual)

Road trip format (Math. models)

Kilometers travelled

Away

Average

Kilometers travelled

Away

Average

Argentino Quilmes Peñarol Bahía Basket Gimnasia Indalo Boca Juniors Obras Sanitarias Lanús La Unión Regatas San Martin Quimsa Olimpíco Atenas Libertad Sionista Estudiantes Instituto Ferro San Lorenzo

24,613 29,026 29,026 31,451 59,947 22,622 22,622 22,622 33,447 24,362 28,293 29,129 29,129 25,807 23,170 23,365 25,247 25,807 22,622 22,622

28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28

879.04 1036.64 1036.64 1123.25 2140.96 807.93 807.93 807.93 1194.54 870.07 1010.46 1040.32 1040.32 921.68 827.50 834.46 901.68 921.68 807.93 807.93

20,014 22,764 22,764 20,372 43,215 18,332 18,332 18,814 26,362 21,626 22,752 20,722 21,397 21,790 18,685 17,967 18,818 20,390 18,992 18,332

28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28

714.79 813.00 813.00 727.57 1543.39 654.71 654.71 671.93 941.50 772.36 812.57 740.07 764.18 778.21 667.32 641.68 672.07 728.21 678.29 654.71

Total

554,929

990.94

432,440

The Second Division team displaying the most significant savings was Hispano Americano, whose home base is Río Gallegos in Argentina’s far south. For that team’s players, a game against Platense in Buenos Aires, for example, means a round trip of more than 50 0 0 kilometers. As is shown in Tables 6 and 7, the mathematical models reduced Hispano Americano’s average travel per away game by 48% in 2014–2015 and 53% in 2015–2016. Under the old format, the team in 2015–2016 would have clocked more than 71,0 0 0 kilometers, greater than a complete trip around the Earth. As it turned out, they won the 2015–2016 championship and were promoted to the First Division. A comparison of the Second Division results with those of the First Division reveals that the travel reductions for the former were particularly impressive. This was so because its previous format of one game per week was even more prejudicial in terms of travel distances than the team-pair setup used by the First Division.

772.21

If we now express the travel reductions in monetary terms, the global financial savings assuming an estimated travel cost of US$ 2 per kilometer (based on current intercity bus fares) for the 2015– 2016 season would have been about US$ 245,0 0 0 (on a reduction of 122,489 kilometers; see Table 5) for the First Division and more than US$ 486,0 0 0 (on a reduction of 243,340 kilometers; see Table 7) for the Second Division. The savings for the two divisions combined would thus have been approximately US$ 731,0 0 0. Note that this is a lower bound since in some cases the teams travelled by airplane, which is considerably more expensive. As can be appreciated from the foregoing analysis, the main source of the savings was the switch to a schedule format with road trips, as is practised in the NBA. Whereas the old formats could be designed with manual methods, the new setup could not possibly have been implemented without the support of the mathematical programming tools described in this study.

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Table 6 Comparison of distances travelled by Second Division teams in 2014–2015 using road trip format (mathematical models) and once a week format (manual methods). Team

Season 2014–2015 Once a week format (Manual)

South 9 de Julio Anzorena Alianza San Luis Banda Norte Deportivo Viedma Estudiantes de Olavarría Ferro Hispano Americano Huracán de Trelew Monte Hermoso San Lorenzo Sport Club Cañadense North Tomás de Rocamora Atlético Echague Instituto de Córdoba Barrio Parque La Unión de Colón Oberá TC Club Atlético San Isidro UNCAus Unión de Santa Fe Villa Ángela Basket Sarmiento Tiro Federal Morteros Total (Both conferences)

Road trip format (Math. models)

Kilometers travelled

Away

Average

Kilometers travelled

Away

Average

20,522 26,840 20,804 18,306 25,880 19,974 20,264 58,918 31,006 25,342 20,264 21,364

14 14 14 14 14 14 14 14 14 14 14 14

1465.86 1917.14 1486.00 1307.57 1848.57 1426.71 1447.43 4208.43 2214.71 1810.14 1447.43 1526.00

13,711 20,159 16,180 14,029 18,470 13,705 20,449 36,915 21,385 17,756 15,400 15,712

17 17 17 17 17 17 17 17 17 17 17 17

806.53 1185.82 951.76 825.24 1086.47 806.18 1202.88 2171.47 1257.94 1044.47 905.88 924.24

13,308 9748 13,186 13,186 13,466 22,160 9944 14,736 9740 14,562 14,534 10,846

14 14 14 14 14 14 14 14 14 14 14 14

950.57 696.29 941.86 941.86 961.86 1582.86 710.29 1052.57 695.71 1040.14 1038.14 774.71

8083 7099 9092 9092 8690 12,797 7163 8831 6586 8877 9905 8944

17 17 17 17 17 17 17 17 17 17 17 17

475.47 417.59 534.82 534.82 511.18 752.76 526.12 519.47 387.41 522.18 582.65 421.35

1395.54

329,030

468,900

806.45

Table 7 Comparison of distances travelled by Second Division teams in 2015–2016 using road trip format (mathematical models) and once a week format (manual methods). Team

Season 2015–2016 Once a week format (Manual)

South Deportivo Viedma Estudiantes de Olavarría Hispano Americano Huracán de Trelew Monte Hermoso Ciclista Gimnasia de La Plata Platense Atenas de Carmen de Patagones Tomás de Rocamora Parque Sur Olimpo Petrolero North Atlético Echague Barrio Parque La Unión de Colón Oberá TC Club Atlético San Isidro Unión de Santa Fe Villa Ángela Basket Sarmiento Tiro Federal Morteros Comunicaciones Hindú Salta Basket UNCAus Total (both conferences)

Road trip format (Math. models)

Kilometers travelled

Away

Average

Kilometers travelled

Away

Average

24,956 22,202 71,598 34,382 25,484 22,896 23,722 22,770 24,956 26,670 27,888 23,416 37,048

18 18 18 18 18 18 18 18 18 18 18 18 18

1386.44 1233.44 3977.67 1910.11 1415.78 1272.00 1317.89 1265.00 1386.44 1481.67 1549.33 1300.89 2058.22

17,255 13,163 33,198 21,751 15,241 13,776 16,804 15,576 16,722 15,516 17,235 15,880 21,432

18 18 18 18 18 18 18 18 18 18 18 18 18

958.61 731.28 1844.33 1208.39 846.72 765.33 933.56 865.33 929.00 862.00 957.50 882.22 1190.67

14,628 21,118 20,240 26,950 15,236 14,942 17,624 15,664 16,696 18,966 15,664 34,616 16,532 636,864

18 18 18 18 18 18 18 18 18 18 18 18 18

812.67 1173.22 1124.44 1497.22 846.44 830.11 979.11 870.22 927.56 1053.67 870.22 1923.11 918.44 1360.82

11,097 12,924 12,624 15,341 11,735 10,414 12,515 11,683 12,071 12,442 10,100 16,923 10,106 393,524

18 18 18 18 18 18 18 18 18 18 18 18 18

616.50 718.00 701.33 852.28 651.94 578.56 695.28 649.06 670.61 691.22 561.11 940.17 561.44 840.86

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7. Discussion, conclusions and current work This study has presented the mathematical programming models used in recent years to determine the season schedules for the first and second divisions of professional basketball in Argentina. A central aspect of the new season schedule formats applied by these models is the use of road trips similar to the setup employed by the NBA in the United States. The modelling is broken down into two stages. The first-stage model defines the schedule with approximate days for each team’s game taking into account the possible road trips (preferred and reasonable ones) with the idea of reducing travel distances. This stage is thus a new variation on the TRTTP, which in turn is a less-studied variant of the well-known TTP. We denote this new variation TP-TRTTP (Trip Preferences-Time Relaxed Travelling Tournament Problem). The second-stage model assigns exact days to each game to be played. Both models include a series of constraints reflecting criteria requested either by the Argentine Basketball Club Association, the broadcaster televising the games or the teams themselves. The idea of splitting the decisions in two stages, first defining a base schedule with approximated days and then assigning specific dates to matches, could be useful for other leagues using a time relaxed schedule, such as the leagues of the main professional sports in North America. Possibilities for future work to improve the scheduling process presented in this paper include the following: for moderately sized instances, designing a single model that combines the two stages developed here and incorporates home-away patterns; evaluating the use of different objective functions in both stages such as maximizing stadium attendance using attendance estimates based on the date on which each game is played; and, finally, adding scheduling fairness restrictions such as rest day balance between games across all teams (using for example the approach proposed in Atan & Çavdarog˘ lu (2018)). The implementation of our models brought savings of more than 30% in the teams’ average distance travelled per away game. This led to a decrease in the two divisions’ global team travel costs of more than US$70 0,0 0 0 per season, not to mention the significant reduction in player fatigue. For some teams however, the savings were well above the average due to their location far from the country’s major centres and the full advantage taken by club management of the road trip preferences they submitted to the league to cut travel mileage. In addition to the case of Hispano Americano, mentioned in Section 5, the First Division team Bahía Básket also cut its total average travel considerably in both seasons scheduled using our approach. Club president Juan Ignacio “Pepe” Sánchez, gold medallist with the Argentine’s national team at the 2004 Olympics, has commented that “thanks to the system developed by the OR group, our team has lowered travel time and costs by 40%, which has brought about a major drop in injuries compared to previous years” (INFORMS, 2016). A number of teams have used these savings to finance the higher cost of air travel for long road trips. Under the old scheduling formats, many of these journeys, which in some cases exceed 10 0 0 kilometers one way, were made by bus. Flying has meant a dramatic improvement in the players’ road trip travel experience. Also taking a positive view of the new setup and the use of mathematical scheduling techniques is Emanuel “Manu” Ginóbili, undoubtedly the greatest name in the history of Argentine basketball. “Considering the huge size of the country and the lack of highways and interconnecting flights,” he observed, “something had to be done. The team of mathematicians at the University of Buenos Aires did a great job optimizing travel times for the players and has saved a lot of money for the teams. No doubt some minor adjustments are needed, but I’m very enthusiastic about the changes” (INFORMS, 2016).

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The satisfaction of Argentine Basketball Club Association management with the results of the mathematical scheduling has prompted it to retain the OR group’s services for the following seasons. The schedules designed by the group have been used since the 2014–2015 season. According to Fabián Borro, the organization’s president, “we could not imagine going back to scheduling the basketball seasons without OR support”. Building on the success of this work, new areas of collaboration between the Association and the OR group are being explored. For 2016–2017 and 2017–2018 seasons a model was implemented to determine the First Division referee assignments, a process that heretofore was carried out manually. The assignments now made using mathematical techniques are proving to be more transparent and more compliant with the criteria set for them, and have resulted in significant referee travel savings as well. Acknowledgments The authors are grateful to the AdC and its member clubs for their collaboration with this scheduling project, and especially to AdC Technical Secretary Sergio Guerrero for his unstinting support on a daily basis. We also want to thank Mario Guajardo, Kenneth Rivkin and Richard Weber for their suggestions that improved the final version of this article. The first author was partially financed by ISCI, Chile (CONICYT PIA FB0816; ICM P-05-004-F), ANPCyT PICT grant 2015-2218 (Argentina) and UBACyT grant 20020170100495BA (Argentina). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ejor.2018.12.018. References Atan, T., & Çavdarog˘ lu, B. (2018). Minimization of rest mismatches in round robin tournaments. Computers & Operations Research, 99, 78–89. Bao, R. (2009). Time relaxed round robin tournament and the NBA scheduling problem. Ph.D. thesis. Cleveland State University. Bean, J., & Birge, J. (1980). Reducing travelling costs and player fatigue in the National Basketball Association. Interfaces, 10, 98–102. Bhattacharyya, R. (2016). Complexity of the unconstrained traveling tournament problem. Operations Research Letters, 44(5), 649–654. Bonomo, F., Cardemil, A., Durán, G., Marenco, J., & Sabán, D. (2012). An application of the traveling tournament problem: The Argentine volleyball league. Interfaces, 42, 245–259. Brandao, F., & Pedroso, J. P. (2014). A complete search method for the relaxed traveling tournament problem. EURO J Comput Optim, 2, 77–86. Campbell, R. T., & Chen, D. (1976). A minimum distance basketball scheduling problem. Management science in sports, 4, 15–26. Costa, D. (1995). An evolutionary tabu search algorithm and the NHL scheduling problem. INFOR: Information Systems and Operational Research, 33(3), 161–178. Craig, S., While, L., & Barone, L. (2009). Scheduling for the national hockey league using a multi-objective evolutionary algorithm. In Proceedings of the Australasian joint conference on artificial intelligence (pp. 381–390). Springer. Durán, G., Guajardo, M., Miranda, J., Sauré, D., Souyris, S., Weintraub, A., & Wolf, R. (2007). Scheduling the Chilean soccer league by integer programming. Interfaces, 37, 539–552. Durán, G., Guajardo, M., & Sauré, J. M. D. (2017). Scheduling the South American qualifiers to the 2018 FIFA world cup by integer programming. European Journal of Operational Research, 262(3), 1109–1115. Easton, K., Nemhauser, G., & Trick, M. (2001). The traveling tournament problem description and benchmarks. In Principles and practice of constraint programming–CP 2001 (pp. 580–584). Springer. Easton, K., Nemhauser, G., & Trick, M. (2003). Solving the travelling tournament problem: A combined integer programming and constraint programming approach. In E. Burke, & P. Causmaecker (Eds.), Practice and theory of automated timetabling IV. In Lecture Notes in Computer Science: 2740 (pp. 100–109). Springer. Fleurent, C., & Ferland, J. (1993). Allocating games for the NHL using integer programming. Operations Research, 41, 649–654. Goossens, D., & Spieksma, F. (2009). Scheduling the Belgian soccer league. Interfaces, 39, 109–118. INFORMS (2016). O.R. transforms scheduling of Chilean soccer leagues and South American World Cup qualifiers. Accessed May 31, 2016 https://www.pathlms. com/informs/events/533/thumbnail_video_presentations/26171.

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