Sports tournament scheduling to determine the required number of venues subject to the minimum timeslots under given formats

Computers & Industrial Engineering 65 (2013) 226–232

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Sports tournament scheduling to determine the required number of venues subject to the minimum timeslots under given formats Ling-Huey Su a,⇑, Yufang Chiu a, T.C.E. Cheng b a b

Department of Industrial and System Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan, ROC Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e

i n f o

Article history: Received 4 January 2012 Received in revised form 23 February 2013 Accepted 26 February 2013 Available online 14 March 2013 Keywords: Sports scheduling Round robin tournament Single elimination Double elimination

a b s t r a c t We studied a two-phase, preliminary and finals, tournament, which commonly adopted for non-professional sports. The round robin tournament in divisions is played in the preliminary phase, followed by one of the three variants, namely single elimination, double elimination, and round robin in the finals phase. The objective is to determine the required number of venues (tables or courts) subject to the least timeslots under the given format. We used a diagonal symmetric matrix to pair teams to games and to schedule games in timeslots for the round robin tournament. For the preliminary phase, we proposed a procedure to find the number of divisions and the number of teams in each division that minimize the total number of games and timeslots accordingly. For the finals phase, we determined the number of venues required in the least timeslots. We then formulated a constraint programming model based on the diagonal symmetric matrix for the round robin tournament. Finally, we provided suggestions for choosing the appropriate competition format. Ó 2013 Published by Elsevier Ltd.

1. Introduction The study of sports tournament scheduling has received a great deal of interest in the operations research literature over the past 30 years due to its wide applications. Most of the literature on sports tournament scheduling has focused on the round robin tournament play. A round robin tournament is a tournament where all the teams meet all the other teams a fixed number of times. The circle method is the starting point for the construction of the single round robin tournament. More recent research focuses on seeking a good schedule to meet various criteria. De Werra (1982) considers the use of oriented factorizations of complete graphs to obtain a schedule that minimizes the number of breaks and the number of days with breaks in the sequences of home games and away games. Knust and Lücking (2009) develop a feasible schedule with the minimum number of breaks and minimum total costs, where place constraints are also taken into account. Briskorn and Drexl (2009) consider a problem with costs associated with each possible match. Kendall, Knust, Ribeiro, and Urrutia (2010) present a review of sports scheduling and the methodologies that have been used.

⇑ Corresponding author. Tel.: +886 3 2654408; fax: +886 3 2654499. E-mail addresses: [email protected] (L.-H. Su), (Y. Chiu), [email protected] (T.C.E. Cheng).

[email protected]

0360-8352/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.cie.2013.02.021

For the double round robin tournament, Della Croce and Oliveri (2006) use integer programming to solve the Italian football league. Rasmussen and Trick (2007) design a hybrid algorithm including integer programming and constraint programming for minimizing breaks. They consider both mirrored and non-mirrored schedules with and without place constraints. Rasmussen and Trick (2008) later survey break minimization and distance minimization in tournaments in the past 30 years. Knust (2010) considers the sports league scheduling problem that is related to scheduling games in non-professional table-tennis leagues. For the triple round robin tournament, Rasmussen (2008) uses the logic-based Benders decomposition and column generation techniques to find the home-away pattern sets and timetables for the Danish professional soccer league. In the two-phase method in which the first phase schedules teams without considering home and away requirements, and the second phase designates the home and away teams, Trick (2001) and Russell and Urban (2006) solve the problem. Henz, Muller, and Thiel (2004) analyze the use of the global constraints for constraint-based search. They find that arc-consistent propagation for the all-different constraint is crucial for the efficient solution of the tournament scheduling problem. Drexl and Knust (2007) give a comprehensive survey of graph-theoretical concepts and decomposition techniques for sports league scheduling. For tournaments in which the venues are unassociated with any of the participating parties, Urban and Russell (2003) develop an integer goal program for scheduling under varying conditions. Ryvkin

L.-H. Su et al. / Computers & Industrial Engineering 65 (2013) 226–232

(2010) compares the selection efficiency of contests, binary elimination tournaments, and round robin tournaments under different conditions, including distribution of player quality, number of players, and noise level. As to the non-professional sports tournament, Li, Liang, Chen, and Morita (2008) use data envelopment analysis (DEA) to evaluate and compare the performance of nations competing in the Olympic Games. They extend DEA by incorporating multiple sets of nation-specific assurance regions for fair measuring and benchmarking the performance of nations at six summer Olympic Games. No works, however, have addressed the problem under study in this paper, which commonly occurs in non-professional sports tournaments such as table tennis and volleyball in Taiwan. A given number of n teams must be paired in a two-phase tournament. In the preliminary, the n teams are partitioned into divisions in which the round robin tournament is played. From each division, some top teams proceed to the finals, where single elimination, double elimination, or round robin is played. Each game must be played in an available venue that is unassociated with either team. The objective is to find the number of venues required in the least timeslots under the given format. The rest of this paper was organized as follows: In Section 2 we introduced the symmetric diagonal matrix to generate a round robin tournament schedule. In Sections 3–5, we presented the first, second, and third variant of the two-phase sports scheduling problem, respectively. In Section 6 we gave the features of the three variants and, finally, in Section 7 we concluded the paper and proposed topics for future research.

The circle method is the starting point on the construction of the single round robin tournament. The merit of diagonal symmetric matrix method is that multiple feasible solutions can be generated compared to only one solution is generated by the circle method, and therefore seeking of better schedule to meet other criteria such as minimum carry-over effect is available. The steps to construct our dispatching rule for an even number of teams in a round robin tournament are as follows: Step 1. Using Constraint Programming to construct a diagonal symmetric matrix wherein all elements in each column are distinct numbers 0, 1, 2, . . . , n  1, and so are elements in each row. Step 2. Assign team i in the column and team j in the row that have element xij in the matrix equal to k to play in the timeslot k, for k = 1, 2, . . . , n  1. Step 3. Thus all teams are assigned in each slot and the tournament schedule can be constructed. Example 1. For a six-team tournament, we first construct the diagonal symmetric matrix in Step 1 (Table 1). Following Steps 2, the tournament is held in slots 1, 2, 3, 4, and 5. The pairs in slot 1 are teams 1 and 2, teams 3 and 4, and teams 5 and 6. Similarly, in slots 2, 3, 4, and 5, teams are paired to play. This arrangement ensures that no team competes with the same team twice. For an odd number of teams in a tournament, the schedule can be constructed in exactly the same way using a (n + 1)  (n + 1) diagonal symmetric matrix and then set the last team as the default. Any team meet the default in a particular slot is a bye. Constraint Programming efficiently generates a diagonal symmetric matrix in which the diagonal elements are zero and all

2. Relationship between round robin tournament and diagonal symmetric matrix In a single round robin tournament, each team plays with each other team exactly once. The meeting of two teams is called a game and the games must be allocated to a number of timeslots (slots). When scheduling a single round robin tournament, at least (n  1) slots are required when the number of teams n is even, and at least n slots are required when n is odd. A round robin tournament with an odd number of teams consists of n slots, in each of which n  1 teams play and one team rests. This team is said to have a bye. The tournament is compact if the number of available slots equals the lower bound and the tournament is relaxed if the number exceeds the lower bound. In a compact round robin with an even number of teams, all the teams play in each slot without a bye. We define a diagonal symmetric matrix to construct a round robin tournament play. An n  n diagonal symmetric matrix is a matrix wherein all elements in each column are distinct numbers 0, 1, 2, . . . , n  1, so are elements in each row. To begin with, the elements in the first column and the first row are in sequential order from 0 to (n  1). All diagonal elements are zero. In addition, all elements are symmetric with respect to the diagonal, i.e., xij = xji, for i, j = 1, . . . , n  1. There is at least one n  n diagonal symmetric matrices exist if n is an even integer. A 6  6 diagonal symmetric matrix is illustrated in Table 1.

Table 1 A 6  6 diagonal symmetric matrix.

1 2 3 4 5 6

1

2

3

4

5

6

0 1 2 3 4 5

1 0 4 5 3 2

2 4 0 1 5 3

3 5 1 0 2 4

4 3 5 2 0 1

5 2 3 4 1 0

227

Fig. 1. The CPL model for the round robin tournament.

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L.-H. Su et al. / Computers & Industrial Engineering 65 (2013) 226–232

3.1. Division round robin tournament in the preliminary phase of DRSE

Table 2 A 6  6 diagonal symmetric matrix.

1 2 3 4 5 6

1

2

3

4

5

6

0 1 2 3 4 5

1 0 5 2 3 4

2 5 0 4 1 3

3 2 4 0 5 1

4 3 1 5 0 2

5 4 3 1 2 0

Table 3 Solution for round robin tournaments obtained by CPL model. n

# Constraints

# Variables

Req’d memory

# Failures in the search tree

Time (s)

8 12 16 20 30 40 50 60 70 80

514 2357 7348 18,015 93,035 298730 737600 1542145 2874865 4928260

128 288 512 800 1800 3200 5000 7200 9800 12,800

972,616 1,112,296 3,041,896 7,082,092 34,920,764 1,10,437,172 2,70,509,748 5,62,963,112 10,47,102,156 17,92,451,308

0 0 0 0 4 0 31 358 17 588

0.00 0.00 0.03 0.06 0.41 1.44 4.22 13.64 17.94 583.34

products in each row and column are distinct. The former part of the OPL model in Fig. 1 is coded for this purpose. In Table 1, if we fix the element in the first row and second column as well as the element in the third row and the fifth column in Table 1 to be the value of 1, then the Constraint Programming will generate another diagonal symmetric matrix as Table 2 shows. By the similar manners, multiple distinct schedules can be generated. We coded the algorithm in OPL Studio 3.5.1 and ran it on a PC with AMD 2.91 GHz. The model can solve problems up to n = 84. Table 3 presents the computational results. The remainder of the discussion uses the following notation. g, m: number of divisions and number of teams in each division in the division round robin of the preliminary phase, respectively. C: total number of games in both the preliminary and finals phases. P: number of slots per day.

Lemma 1. In the DRSE tournament format in which two teams in each division in the preliminary proceed to the finals, the smallest number of teams in each division gives the minimum number of total games. Proof. In the preliminary phase, the number of games in each division is m(m  1)/2 and the number of teams eliminated in each division is (m  2); therefore, the number of games for elimination of one team is [m(m  1)/2]/(m  2). h If the number of teams in each division is now devoted as m0 with m0 > m, then the average number of games for elimination of one team will be [m0 (m0  1)/2]/(m0  2). The difference of the ‘‘elimination efficiency’’ will be

½m0 ðm0  1Þ=2=ðm0  2Þ  ½mðm  1Þ=2=ðm  2Þ ¼ ½ðm0 m0 m  2m0 m0  m0 m þ 2m0 Þ  ðmmm0  2mm  mm0 þ 2mÞ=½2ðm  2Þðm0  2Þ ¼ fðm0  mÞ½m0 m  2ðm0 þ mÞ þ 2g=½2ðm  2Þðm0  2Þ ¼ fðm0  mÞ½ðm0  2Þðm  2Þ  2g=½2ðm  2Þðm0  2Þ Since the smallest number of teams in each division requiring a nontrivial analysis is three for the division round robin tournament, thus m P 3 and m0 P 4, implying {(m0  m)[(m0  2) (m  2)  2]}/[2(m  2)(m0  2)] > 0, that is, it takes more games to eliminate teams in the preliminary if m0 > m. In the final phase, it takes one game to eliminate one team; therefore, the number of games is minimized when the number of teams in each division is minimized or when the number of divisions is maximized. According to diagonal symmetric matrix in Section 2, m = 3 and m = 4 incur the same (also minimum) number of slots. Thus, m = 4 is selected since it requires no bye and incurs fewer number of divisions compared with m = 3. When n is not divisible by 4, at least one division includes three teams. Let a and b be the number of divisions with four and three teams in the division round robin, respectively, then

b ¼ 0; 1; 2; 3 if ðn mod 4Þ ¼ 0; 3; 2; 1; a ¼ ðn  3  bÞ=4; g ¼aþb

ð1Þ

3. Division round robin tournament in the preliminary and single elimination in the finals (DRSE)

For example, if n = 41, then a = 8 and b = 3, and the number of divisions is g = 8 + 3 = 11.

The first variant is division round robin in the preliminary and single elimination in the finals (DRSE). In a single elimination tournament, the loser of each game is immediately eliminated from the tournament. Since the structures of single elimination is identical to the winner’s bracket of double elimination, therefore, for an overview of the single eliminations, readers can refer to the winner’s bracket of double elimination in Appendix A or the discussion in http://www.printyourbrackets.com/Tennisbrackets.html. A number of n teams are partitioned into divisions and many top teams from each division proceed to the finals. In our case, we chose two top teams to be promoted since it is the most common case in the table tennis or volley ball plays in Taiwan. For the problem with different number of teams being promoted, the analysis is similar. Each game is played at one of the available venues. The solution procedure for the DRSE problem is divided into two stages. The first stage determines the number of divisions and the number of teams within each division that minimizes the total number of games in both the preliminary phase and the finals. The second stage determines the number of venues required.

3.2. Venues required for DRSE The single elimination tournament in the finals has 2g teams, 2k1 6 2g 6 2k and k = (log22g). If 2g is not equal to a power of 2, 2k1 < 2g < 2k, then a number of teams 2(2g  2k1) teams appears in the first slot to enable 2k1 teams to play in the second slot. Let T(r) be the number of teams playing in the rth slot of the single elimination tournament, i.e.,

TðrÞ ¼ ð2g  2k1 Þ  2 ¼ 4g  2k ; TðrÞ ¼ 2

kðr1Þ

;

r¼1

r>1

If 2g is equal to a power of 2, then TðrÞ ¼ 2kðr1Þ ; r P 1. In both cases, when r = k, the tournament is set up. Therefore, the single elimination includes k slots. In DRSE, a bottleneck in venues occurs in the first or second slot of the finals. Let C(r) be the number of games in the rth slot of the finals and V be the number of venues required. Noting CðrÞ ¼ TðrÞ , 2 we consider two cases as follows:

L.-H. Su et al. / Computers & Industrial Engineering 65 (2013) 226–232

(5) (1)

(7)

Tw ðrÞ ¼ 4g  2k ;

r¼1

ð2Þ

T w ðrÞ ¼ 2kðr1Þ ;

r–1

ð3Þ

r¼3

ð4Þ

k

Tl ðrÞ ¼ 4g  2 ; (3)

kb2r c

T l ðrÞ ¼ 2 (9)

(2)

Winner

(6)

r > 3:

;

(4)

T w ðrÞ ¼ 2

kðr1Þ

;

Tl ðrÞ ¼ 4g  3  2

Fig. 2. Movement of one game in the second round to the first round.

(i) Cð1Þ P Cð2Þ, as in the 14-team diagram depicted in Appendix A. Since the games in the first slot should be played before those in the second slot, V = max{C(1), C(2)} = C(1), where C(1) = (4g  2k)/2 = 2g  2k1 and C(2) = 2k1/2 = 2k2. (ii) C(1) < C(2), as in the 10-team diagram depicted in Appendix A. Since some games in the second slot have no relationships with those in the first slot and can thus be moved to the first slot to reduce the number of venues required (see Fig. 2), we have

j

C 0 ð1Þ ¼ 2g  2k1 þ Cð2ÞCð1Þ 2 j k 0 k2 Cð2ÞCð1Þþ1 C ð2Þ ¼ 2  2

k

V ¼ max½C 0 ð1Þ; C 0 ð2Þ 4. Division round robin tournament in the preliminary and double elimination in the finals (DRDE) A double elimination tournament is a type of elimination tournament in which a team ceases to be eligible to play in the tournament upon having lost two games. The tournament is broken into sets of brackets, the Winners (W) Bracket and Losers (L) Bracket as shown in Appendix A. 4.1. Division round robin tournament in the preliminary phase of DRDE The proof that shows the number of games is minimized when the number of teams in each division is minimized or when the number of divisions is maximized in the preliminary for DRDE is the same as that of DRSE. In the final phase, it takes two games to eliminate one team; therefore, the number of games is minimized when the number of teams in each division is minimized or when the number of divisions is maximized. Since the nontrivial analysis m = 3 requires the same (also minimum) number of rounds as that of m = 4, we select m = 4 because the required number of divisions is minimized. If the total games n is not divisible by 4, we apply Eq. (1).

ð5Þ

Case 2: If 2g  2k1 > 2k2, then

T w ðrÞ ¼ 4g  2k ; (8)

229

T l ðrÞ ¼ 2kb2c ; r

r¼1

ð6Þ

r–1

ð7Þ

k1

;

r¼2

r > 2:

ð8Þ ð9Þ

Eq. (2) is calculated as follows: Let X be the number of teams with a bye in the opening slot. To enable 2k1 teams to play in Tw(2), we set 12[2g  X] = 2k1, which implies X = 2k  2g. Therefore, Tw(1) = 2g  X = 4g  2k. Eq. (3) states that 2k1 teams play in Tw(2). The number of teams is then halved in the following slots. In the L Bracket, the first game is played in the third slot because it follows the first (opening) and second slots of the W Bracket. Eq. (4) is calculated as follows: The number of teams with a bye in the first slot of the L Bracket (Tl(3)) is 2k1  ½12 ð2g  X þ 2k2  ¼ 2k  2k2  2g. Thus Tl(3) = 2g  2k1 + 2k1  X = 4g  2k. Both the losers in the W Bracket and the winners in the L Bracket then proceed to the successive odd slot of the L Bracket while only the winners in the L Bracket proceed to the successive even slot of the L Bracket (Eq. (5)). In Case 2 where 2g  2k1 > 2k2, the calculations of Eqs. (6) and (7) for the W Bracket are similar to those of Eq. (2) and Eq. (3), respectively. Eq. (8) is calculated as follows: In the L Bracket, the first play occurs in the second slot because it only follows the first (opening) slot of the W Bracket. Let X be the number of teams with a bye in the second slot of the L Bracket. To enable 2k1 teams to play in Tl(3), we set [2g  2k1  X]/2 + X + 2k2 = 2k1, which implies X = 2k  2g. Therefore, Tl(2) = [(2g  2k1)  X] = (2g  2k1)  (2k  2g)= 4g  3  2k1 Eq. (9) is calculated similarly to Eq. (5). r Since T l ðrÞ ¼ 2kb2c ; r > 2, the tournament requires k br=2c ¼ 0, implying r = 2k slots to determine the champion of the L Bracket. However, eliminating the champion of the W Bracket after only one loss would be unfair. Therefore, the L Bracket champion must beat the W Bracket champion twice. Thus, the total number of slots to produce a final champion is 2k + 1. If the number of teams equals a power of 2, the numbers of teams in the W Bracket and L Bracket are

T w ðrÞ ¼ 2kðr1Þ ; kbr=2c

T l ðrÞ ¼ 2

;

r P 1; r P 1:

Therefore, the total slots needed to determine the final champion is 2k + 1. 4.3. Scheduling venues

4.2. Double elimination tournament in the finals If the number of teams remaining after the preliminary phase 2g is not equal to a power of 2, i.e., 2k1 < 2g < 2k, then a number of teams 2(2g  2k1) appear in the opening slot. The two possibilities in the first slot of the L Bracket are (2g  2k1) 6 2k2 and (2g  2k1) > 2k2 as the 10- and 14-teams tournament brackets shown in Appendix A, respectively. Let Tw(r) and Tl(r) be the number of teams in the rth slot of the W Bracket and L Bracket, respectively. Case 1: If 2g  2k1 6 2k2 , then

The next step is to determine the number of venues required. In DRDE, the bottleneck in venues appears in one of the first three rounds in the finals. Let V(r) denote the number of venues required in the rth slot of the finals and let V = max [V(1), V(2), V(3)]. If n is a power of 2, then V = max [V(1), V(2), V(3)] = V(1) = V(2) = 2k1, where k = (log22g). If n is not a power of 2, two cases may occur. Case 1. If 2g  2k1 > 2k2, then V = max [V(1), V(2), V(3)] = V(1) = V(2) = (2g  2k1)/2 = g  2k2, as in the 14-team tournament depicted in Appendix A.

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L.-H. Su et al. / Computers & Industrial Engineering 65 (2013) 226–232

Case 2. If 2g  2k1 6 2k2 , then V = max [V(1), V(2), V(3)] = max [V(2), V(3)], where V(2) = 2k1 and V(3) = [Tw(3) + Tl(3)]/ 2 = 2g  3  2k3. Let DV = V(3)  V(2) = 2g  5  2k3. Then (i) If DV 6 0, then V = V(2) = 2k2, as in the 10-team tournament depicted in Appendix A. k2 k Þ l ð3Þ (ii) If DV > 0, then V ¼ Vð3Þ ¼ T w ð3ÞþT ¼ 2 þð4g2 ¼ 2g  3 2 2 2k3 , as in the 14-team tournament shown in Appendix A. 5. Division round robin tournament in the preliminary followed by round robin tournament in the finals (DRR) 5.1. Division round robin tournament in the preliminary phase of DRR Like DRSE, the number of divisions and the number of teams in each division is determined. The top two teams in the preliminary advance to the finals, so the total number of games C in DRR is 2g C ¼ Cm 2  g þ C2 ¼

mðm  1Þ ð2gÞð2g  1Þ gþ 2 2

Assume that g and m are reversed such that m and g denote the number of divisions and the number of teams in each division, respectively. The total number of games C0 is then

C 0 ¼ C g2  m þ C 2m 2 ¼

gðg  1Þ ð2mÞð2m  1Þ mþ 2 2

games and the number of required venues are both 2g ¼ g. Recall 2 that the diagonal symmetric matrix introduced in Section 2 for the round robin tournament ensures the satisfaction of the following constraints: (i) Each team plays each other team once. Each team plays in each round no more than once. (ii) If the venues must be scheduled, an additional constraint is: (iii) No more than one team can compete in the same venue in the same round. The OPL implementation of pairing the teams for games and scheduling the games for venues in the round robin tournament is shown in Fig. 1. 6. Comparison of the three formats We compared the three competition formats, namely DRSE, DRDE and DRR, in terms of total number of games and total number of rounds required. 6.1. Comparison of total number of games The total number of games for (i) DRSE is

Cm 2 g þ 2g  1

S ¼ C  C0 mg½ðm  1Þ  ðg  1Þ ½ð2gÞð2g  1Þ  ð2mÞð2m  1Þ þ 2 2 ½m  g½mg  4m  4g þ 2 ¼ 2 ¼

The values of S based on various values of m and g are shown in Appendix B, which can be classified into two cases: (1) If m > g and (i) g 6 4; then C < C0 , implying that C is minimized when the number of teams in each divisions is minimized or when the number of divisions is maximized. (ii) g P 8, then C0 < C, implying that C0 is minimized when the number of teams in each division is minimized or when the number of divisions is maximized. By induction, the number of total games is minimized when the number of teams in each division is the minimum or when the number of divisions is the maximum. Since m > g, the optimum appears when the difference between m and g is smallest. (2) m < g and (i) m 6 4; then C0 < C, implying that C0 is minimized when the number of teams in each division is maximized or when the number of divisions is minimized. (ii) m P 8; then C < C 0 , implying that C is minimized when the number of teams in each division is maximized or when the number of divisions is minimized. By induction, the number of total games is minimized when the number of teams in each division is the maximum or when the number of divisions is the minimum. Since m < g, the optimum is reached when the difference in the values of m and g is smallest. Generally, the total games and also the total slots are optimum when the difference between the values m and g is minimized.

¼ mðm1Þ g þ 2g  1 2 (ii) DRDE is

Cm 2 g þ 2  2g  1 ¼ mðm1Þ g þ 4g  1 2 (iii) DRR is

¼

2g Cm 2  g þ C2 mðm1Þ ð2gÞð2g1Þ gþ 2 2

The total number of games of DRSE is evidently the smallest among the three formats. The next comparison is between ð2gÞð2g1Þ ¼ 2g 2  g and 4g  1 for DRR and DRDE, respectively. Since 2 the smallest number of divisions in the division round robin tournament requiring a nontrivial analysis is two, we consider the following two cases: (i) g = 2 since 2g2  g  (4g  1) < 0, so DRSE < DRR < DRDE. (ii) g > 2 since 2g2  g  (4g  1) > 0, so DRSE < DRDE < DRR. 6.2. Comparisons of total number of rounds required Since all three competition formats involve a division round robin tournament, we compare their total numbers of rounds in the finals. The total number of rounds required in the finals for (i) DRSE is

P f ¼ dlog2 2ge: (ii) DRDE is

P f ¼ 2dlog2 2ge þ 1 (iii) DRR is

P f ¼ 2g  1

5.2. Scheduling venues For DRR, the venue bottleneck requirement occurs in the finals. When 2g teams advance to the finals, the number of required

The total number of rounds of DRSE is evidently the smallest. We then compare 2dlog2 2ge þ 1 with 2g  1 for DRDE and DRR, respectively. There are two possible outcomes:

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L.-H. Su et al. / Computers & Industrial Engineering 65 (2013) 226–232

(i) g 6 5: Since 2dlog2 2geþ 1  2g þ 1 P 0, so DRSE < DRR < DRDE. (ii) g > 5: Since 2dlog2 2ge þ 1  2g þ 1 < 0, so DRSE < DRDE < DRR. 7. Conclusion We considered the scheduling problems associated with the three variants of the two-phase sports competition, where there is a division round robin tournament in the preliminary phase, followed by single elimination, double elimination, or round robin tournament in the finals. The objective is to determine the required number of venues subject to the minimum timeslots under a given format. This tournament commonly occurs in non-professional sports tournaments such as table tennis and volleyball play in Taiwan. We used the diagonal symmetric matrix to generate the round robin tournament and use constraint programming to construct it. The merit of this method is that multiple feasible solutions can be generated comparing to only one solution generated

by the previous method, and therefore seeking of better schedule to meet other criteria such as minimum carry-over effect is available. The model can solve problems involving up to 84 teams for scheduling the round robin tournament within 1 h. We also compared the three competition variants and suggest the use of a proper format for scheduling sports competitions under different conditions. Our results help close the gap between theory and application in sports scheduling. This study is the first to analyze the popular two-phase sports tournament. Future studies may consider a more sophisticated constraint programming model to improve the efficiency of solving sports scheduling problems involving multiple constraints.

Appendix A Double elimination bracket for 10 and 14 teams.

10-team Double Elimination Winner's Bracket (3) (11)

(1) (5)

(15)

(6)

(2)

(12) (4)

(18)

Loser's Bracket L2 L5

L6

(13)

(7) L3 L4

L1

L13

L12

(17)

(9)

(19)

(16) (10)

(8)

(14) L11

14-team Double Elimination Winner's Bracket (1)

(7) (17)

(2) (3) (4) (5) (6)

(8) (23) (9) (18)

Loser's Bracket L8 (15) (11) L6 L7 L1 L10 L9 (12)

(26)

(10) L21

L18 (21) (19)

(25)

(13) (24)

(14) (20) (16)

(22) L17

(28)

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L.-H. Su et al. / Computers & Industrial Engineering 65 (2013) 226–232

Appendix B

Appendix B (continued) m>g

Values of S for varying values of m and g in DRR variant. (1) m > g m

(2) m < g g

S

m

g

3

2

-6

2

3

6

4

2

-14

2

4

14

5

2

-24

2

5

24

4

3

-7

3

4

7

5

3

-15

3

5

15

6

3

-24

3

6

24

5

4

-7

4

5

7

6

4

-14

4

6

14

7

4

-21

4

7

21

6

5

-6

5

6

6

7

5

-11

5

7

11

10

5

-20

5

10

20

17

5

-6

5

17

6

18

5

0

5

18

0

19

5

7

5

19

-7

20

5

15

5

20

-15

10

6

-4

6

10

4

11

6

0

6

11

0

12

6

6

6

12

-6

8

7

-1

7

8

1

9

7

1

7

9

-1

9

8

3

8

9

-3

10

8

10

8

10

-10

11

8

21

8

11

-21

m>g m 3 4 5 4 5 6 5 6 7

S

S

m

2 2 2 3 3 3 4 4 4

6 14 24 7 15 24 7 14 21

2 2 2 3 3 3 4 4 4

g

S 3 4 5 4 5 6 5 6 7

m

g

S

m

g

S

6 7 10 17 18 19 20 10 11 12 8 9 9 10 11

5 5 5 5 5 5 5 6 6 6 7 7 8 8 8

6 11 20 6 0 7 15 4 0 6 1 1 3 10 21

5 5 5 5 5 5 5 6 6 6 7 7 8 8 8

6 7 10 17 18 19 20 10 11 12 8 9 9 10 11

6 11 20 6 0 7 15 4 0 6 1 1 3 10 21

References

m
m
6 14 24 7 15 24 7 14 21

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