extension and solution of the competition scheduling model...

10
Extension and Solution of the Competition Scheduling Model in Iranian Professional Football League K. Shahanaghi * , M. Mohammadi Darani & M. Moshref Javadi Kamran Shahanaghi, Assistant professor of Industrial Eng-Iran University of Science and Technology Milad Mohammadi Darani, Graduate student of Industrial Eng-Iran University of Science and Technology Mohammad Moshref Javadi, Graduate student of Industrial Eng-Iran University of Science and Technology Keywords 1 ABSTRACT Recently, due to growth in sport branches and increase in constraints and complications in sport competitions, mathematical models and operations research approaches have been significantly used for preparing sport schedules .Football leagues are also among such these competitions. In this paper, the scheduling problem of Iranian football league in which the goal is to minimize breaks and carry over effect is investigated. A new criterion, so-called “breaks for most popular teams” is introduced that has never been taken into account in literature. Final schedule will be obtained through three stages with different objective for each stage in which graph theory and simulated annealing algorithm have been used to solve the problem. Finally, comparing the results of the proposed algorithm with those in the literature shows effectiveness of the proposed algorithm. © 2012 IUST Publication, IJIEPM. Vol. 22, No. 4, All Rights Reserved * Corresponding author. Kamran Shahanaghi Email: s[email protected] Februray 2012, Volume 22, Number 4 pp. 383-392 http://IJIEPM.iust.ac.ir/ International Journal of Industrial Engineering & Production Management (2012) Sport Scheduling, Break Minimization, Carry-over effect Minimization, Simulated Annealing Downloaded from ijiepm.iust.ac.ir at 16:36 IRDT on Saturday August 29th 2020

Upload: others

Post on 15-Jul-2020

4 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

Extension and Solution of the Competition Scheduling

Model in Iranian Professional Football League

K. Shahanaghi*, M. Mohammadi Darani & M. Moshref Javadi

Kamran Shahanaghi, Assistant professor of Industrial Eng-Iran University of Science and Technology Milad Mohammadi Darani, Graduate student of Industrial Eng-Iran University of Science and Technology Mohammad Moshref Javadi, Graduate student of Industrial Eng-Iran University of Science and Technology

Keywords 1ABSTRACT

Recently, due to growth in sport branches and increase in constraints

and complications in sport competitions, mathematical models and

operations research approaches have been significantly used for

preparing sport schedules .Football leagues are also among such

these competitions. In this paper, the scheduling problem of Iranian

football league in which the goal is to minimize breaks and carry over

effect is investigated. A new criterion, so-called “breaks for most

popular teams” is introduced that has never been taken into account

in literature. Final schedule will be obtained through three stages

with different objective for each stage in which graph theory and

simulated annealing algorithm have been used to solve the problem.

Finally, comparing the results of the proposed algorithm with those in

the literature shows effectiveness of the proposed algorithm.

© 2012 IUST Publication, IJIEPM. Vol. 22, No. 4, All Rights Reserved

**

Corresponding author. Kamran Shahanaghi Email: [email protected]

FFeebbrruurraayy 22001122,, VVoolluummee 2222,, NNuummbbeerr 44

pppp.. 338833--339922

hhttttpp::////IIJJIIEEPPMM..iiuusstt..aacc..iirr//

IInntteerrnnaattiioonnaall JJoouurrnnaall ooff IInndduussttrriiaall EEnnggiinneeeerriinngg && PPrroodduuccttiioonn MMaannaaggeemmeenntt ((22001122))

Sport Scheduling,

Break Minimization,

Carry-over effect Minimization,

Simulated Annealing

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 2: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

**

-

*

[email protected]

[email protected]

[email protected] - Sport Scheduling

-

-

3 - Combinatorial Mathematics 4 -Rasmussen and Trick 5 - Kendall et al. 6 - De Werra. 7 - Rosa and Wallis

hhttttpp::////IIJJIIEEPPMM..iiuusstt..aacc..iirr//

ISSN: 2008-4870

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 3: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

-

1 - Bartsch et al. 2 - Della Corce and Oliveri 3 - Ribeiro and Urrutia 4 - Rasmussen 5 - Bender Decomposition

-

n

l>=1

((n*(n-1))/2)l

(n-1)l

ij

t=1,…,(n-1)l

l=1

l=2

.

n*(n-1) ,..., \itopp 1 n i

it

6 - Time-Constrained 7 - round 8 - Compact 9 - Time-Relaxed 10 - Single round-robin tournament (SRRT) 11 - Double round-robin tournament (DRRT) 12 - Mirrored Schedule 13 - Opponent schedule 14 - Pattern set 15 - Home-Away pattern

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 4: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

n*(n-1)

it

t=2,…,n-1

ihi,t-1,hiti

t

-

1 - break

nSRRT

n-2-

3n-6

SRRTn

ij

itjt+1

t1n

xyij

xyj

i

n*ncij

ijcii=0

n n

ij ij

j 1 i 1

c c n 1

i,j = 1,…,n

coen n

2ij

i 1 j 1

coe c

coen(n-1)

i

t

-

t+1j

2 - Mirrored double round-robin tournament 3 - carry-over effect

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 5: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

ji

n

Cijtitj

N={1,…,n}

t=1,…,T=n-1

xijtitj

britit

cijtij

t

SRRT

1 - Russell

xijt

brit

n-2

n-

1

nk=1,…,n

ij

( ) ( )

\{ , }

( ) ( )

n 1

ij ilk lik jl k 1 lj k 1

l N i j k 1

co x x x x

min

n n2ij

i 1 j 1

co

-

{ }

min

T

ijt ijt

i N j N i t 1

c x

Subject to:

( ) , , ,

T

ijt jit

t 1

x x 1 i j N i j

\{ }

( ) , , ,...,ijt jit

j N i

x x 1 i N t 1 T

\{ }

( ) , , ,...,ijt 1 ijt it

j N i

x x br 1 i N t 2 T

\{ }

( ) , , ,...,jit 1 jit it

j N i

x x br 1 i N t 2 T

T

it

i N t 2

br n 2

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 6: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

t=2,…,2n-2i

,i s p \j N iTk

Tkk

k N

bri

{ , }

min i

i s p

br

NP-hard

-

ij

.

Fii

n

( , ) ( ( , ), ( , )) : ,..., ,

,...,

inF n i a i k b i k k 1 1

2

i 1 n 1

1 - Constraint Programming 2 - Placeholder

( )

\{ } \{ }

( )

\{ } \{ }

( )

\{ } \{ }

( )( )

( )( )

( )( )

ik t 1 kit k

k N i k N i2n 2

i ki t 1 k kit

k 2 k N i k N i

ik t 1 ikt

k N i k N i

x x T

br x T x

x x

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 7: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

( )a i k = i k ( )i k n

i k n 1 ( )i k n

( )b i k = i k ( )i k 0

i k n 1 ( )i k 0

(n,i)nii

i

(a(i,k),b(i,k))a(i,k)k

b(i,k)k

n-2n-1

n-

-

psCps

D

-

F(x,z)

zirri

hrtrt

hrtps

min ( , )F x z

Subject to:

,ir

r N

z 1 i N

,ir

i N

z 1 r N

\{ }

, ; ,...,ijt rt ir

j N i r N

x h z 0 i N t 1 T

, ( , ) ,( , )il imz z i j D l m C

-

SA

1 - Simulate Annealing (SA) 2 - Kirkpatrick et al. 3 - Wright

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 8: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

SA

T0

T

-

Cijt

SA

-

-

1 - Willis and Terrill

-

-

-

2 - Miyashiro and Matsui

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 9: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

n=18,20n=18

-

-

SA

SA

Cijt

[1] Rasmussen, R.V., Trick, M.A., "Round Robin

Scheduling-a Survey", European Journal of Operational

Research, Vol. 188, pp. 617-636, 2006.

[2] Kendall, G., Knust, S., Ribeiro, C.C., Urrutia, S.,

"Scheduling in Sports: an Annotated Bibliography",

Computers & Operations Research, Vol. 37, 2010, pp.

1-19.

[3] de Werra, D., "Scheduling in Sports". In: Hansen, P.

(Ed.), Studies on graphs and discrete programming,

North-Holland, Amsterdam, The Netherlands. 1997,

pp. 381-395.

[4] Rosa, A., Wallis, W.D., “Premature Sets of 1-Factors

or How not to Schedule Round-Robin Tournaments”.

Discrete Applied Mathematics, 4: 1982, 291-297.

[5] de Werra, D.," Some Models of Graphs for Scheduling

Sports Competitions". Discrete Applied Mathematics,

21: 1988, 47-65.

[6] Bartsch, T., Drexl, A., Kroger, S., "Scheduling the

professional. soccer leagues of Austria and Germany",

Computers & Operations Research, Vol. 33, No. 7,

2006, pp. 1907-1937.

[7] Della Croce, F., Oliveri, D.,"Scheduling the Italian

football League: an ILP-based Approach ", Computers

& Operations Research, Vol. 33, No. 7, 2006, pp.

1963-1974.

[8] Ribeiro, C.C., Urrutia, S., "Scheduling the Brazilian

Soccer Tournament with Fairness and Broadcast

Objectives", Lecture notes in computer science, Vol.

3867, 2007, pp. 149-159.

[9] Durán, G., Guajardo, M., Miranda, J., Sauré, D.,

Souyris, S., Weintraub, A., Wolf, R., "Scheduling the

Chilean Soccer League by Integer Programming ",

Interfaces, Vol. 37, No. 6, 2007, pp. 539-552.

[10] Rasmussen, R.V., "Scheduing a Triple Round Robin

Trournament for the Best Danish Soccer League",

European Journal of Operational Research, Vol. 185,

2008, pp. 795-810.

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020

Page 10: Extension and Solution of the Competition Scheduling Model ...ijiepm.iust.ac.ir/article-1-808-fa.pdf · زمانبندی ورزشی; حداقل کردن وقفه; حداقل کردن

[11] Russell, K., "Balancing Carry-Over Effects in Round

Robin Tournaments". Biometrika;67: 1980, 127–31.

[12] Knust, S., Lucking, D., "Minimizing Costs in Round

Robin Tournaments with Place Constraints",

Computers & Operations Research, Vol. 36, No. 11,

2009, pp. 2937-2943.

[13] Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.,

"Optimization by Simulated Annealing", Science, Vol.

220, 1983, pp. 671-680.

[14] Wright, M.B., "Scheduling fixtures for basketball New

Zealand", Computers and Operations Research, Vol.

33, No. 7, 2006, pp. 1875-1893.

[15] Willis, R.J., Terrill, B.J., "Scheduling the Australian

State Cricket Season using Simulated Annealing",

Journal of the Operational Research Society, Vol. 45,

1994, pp. 276-280.

[16] Miyashiro, R., Matsui, T., "Minimizing the Carry-Over

Effects Value in a Round Robin Tournament", in: E.

Burke, H. Rudova, (Eds.), Proceedings of the 6th

International Conference on the Practice and Theory of

Automated Timetabling, 2006,pp. 402-405.

Dow

nloa

ded

from

ijie

pm.iu

st.a

c.ir

at 1

6:36

IRD

T o

n S

atur

day

Aug

ust 2

9th

2020