Sports Scheduling Written by Kelly Easton, George - PowerPoint Presentation

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Sports SchedulingWritten by Kelly Easton, George Nemhauser, Michael Trick

Presented by Matthew Lai

Introduction

Paper covers tournament scheduling problems

Single Round Robin Tournament Problem (SRRTP)

Double Round Robin Tournament Problem (DRRTP)

Balanced tournament Design Problem (BTDP)

Bipartite Tournament Problem (BTP)

Traveling Tournament Problem (TTP)

Sports Scheduling Terms

Break

– Number of times that a team has consecutive home/away games

Tournament

– A series of games in which teams play each other team

Schedule

– Mapping of games into slots (time periods) such that each team plays at most once in each slot

Mirrored/Partially Mirrored

– DRRT mirrored such that teams play in same schedule but with mirrored venues

Compact

– Schedule that includes minimum possible number of slots

Pattern

– A vector of home, away, or bye designations for a single team over the slots of a schedule

Complementary

– Two patterns where each slot is complementary to each other

Pattern

Set

– A collection of patterns, one for each team

Tour

– A schedule for a single team in the tournament

Trip

– A series of consecutive away games

Home Stand

– A series of consecutive home games

General Overview – Solutions used in Sports Scheduling

Graph Algorithms

Integer Programming

Constraint Programming

Metaheuristics

Simulated Annealing

Tabu

Search

Single Round Robin Tournament

Formal Definition

Input: A set of

n

teams

T

= { 1, 2, …, n }

Output: A mapping of the games in the set G = {

g

ij

:

i

, j

T,

i

< j}

To slots in the set S = {

s

k

, k = 1, …,

n

-1 if

n

is even and k = 1, …,

n

if n is odd}

SSRT can be described as a Block Design Algorithm

Described as (

v

,

k

,



)

Block Design Algorithm attempts to create subsets of size

k

using the

v

elements of the set where each element can only appear in



subsets

SSRT is therefore (

n

, 2, 1)

SSRTP Circle Method - SSRTP-C

Because the block design for SSRTP is resolvable, circle method can be usedLabel teams , 1, 2, …, n-1. On day i, play i vs. , (i-1) vs (i+1) …, () vs (). Each integer being reduced (mod n – 1) to lie in interval [1, n -1]. Replace  with nExample: n = 8SRRTP-V is a similar algorithm to SRRT-C but includes venue selection so that home/away games for each team are somewhat balanced

Day 1

8

vs 1

7 vs 2

6 vs 3

5 vs 4

Day 2

8 vs 2

1

vs 3

7 vs 4

6 vs 5

Round Robin Tournament and Latin Squares

RRT can reduce to Latin Squares problemsMuch research has been done on Latin Squares and can be adapted to solve RRT problems

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2

3

4

5

6

1

6

2

3

4

5

1

2

2

6

4

5

1

3

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3

4

6

1

2

5

4

4

5

1

6

3

2

5

5

1

2

3

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3

5

2

4

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Round Robin Pattern Sets

Previous methods construct schedules by assigning team matchups to schedule slots. The reverse process has also been suggested to be an effective alternative

Generate pattern set for each team after which compatible teams can be paired

Generating good pattern sets, therefore, is necessary for a good schedule

Round Robin Pattern Sets (cont.)

Patterns

generated for each team will either be feasible patterns, where patterns for the set of teams will generate a RRT, or it they will be infeasible, where patterns cannot be matched to form an

RRT

Whole set of necessary conditions not currently known

Each pair must differ by at least one slot

For DRRT, every pattern pair generated for any two teams must contain, alternatively, a home and away pattern

For all RRT, every slot in pattern set must include equal number of home and away games

Balanced Tournament Designs

Input: A set of

n

teams T = { 1, …,

n

} and a number of facilities F

Output: Mapping of the games in set G = {

g

ij

:

i

,

j

 T,

i

< j } to slots available at each facility as

decribed

by S = {

s

fk

, f = 1, …, F, k = 1, …,

n

– 1 if

n

is even and

n

if

n

is odd } such that no more than one game involving team

i

is assigned to a particular slot and the difference between the number of

apperances

of team

i

at two separate facilities is no more than 1

BTDP-NO

“Bracelet” algorithm – Works with 2m+1 odd teamsArrange teams 1 through 2m + 1 into an elongated pentagon or “bracelet”. Indicate facility associated with each row containing two teams.For each slot k= 1, …, 2m+1 give the team at the top of the pentagon the bye. For each row with two teams, i, j associated with facility f, assign gij to skf. Then shift teams one position in CW directionExample: m = 2

1

2

3

4

5

Facility 1

Facility 2

Bipartite Tournament

Input: Two teams with

n

players T

1

= { x

1

, …,

x

n

} and T

2

 = {y

1

, …,

y

n

}

Output: A mapping of the games in the set

G = {

g

ij

:

i

 T

1

, j  T

2

},

to the slots in set S = {

s

k

, k = 1, …,

n

} such that exactly one game including

t

is mapped to any given slot for all t



T

1

 T

2

In other words, given two groups, ensure that each member of one group plays the members of the other group.

Not just limited to teams of players, but can also encompass Leagues and conferences of teams.

Also equivalent to Latin Square

Graph Algorithm

Sports Scheduling problem can also be solved as a graph problem

The sports schedule would be represented on a graph with 2m teams, as a graph K

2m

with 2m different edge values or colors. Edge [

i

, j] would represent a game between team

i

and team j where

i

and j are vertex nodes.

SRRT

prblem

can be presented as a 1-factorization of K

2m

such that each vertex is not connected by any two edges of the same value. In this way, by taking the 1-factor of a certain value, you receive a perfect matching of pairs.

This graph should be oriented such that the directions indicate the home/away orientation of the game

Graph Example

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6

5

4

Sports Scheduling Problems

Minimum Breaks Problem – minimize number total breaks, home stands and road trips. With even teams, only two feasible schedules exist with no breaks. With odd teams, many more schedules exist

Geographical Location – Ensure that geographically close teams do not play each other consecutively to ensure maximum fan

attendenance

Divisions – Sports leagues often arrange teams into divisions based on geographical closeness. Therefore, matchup inter‑division games on weekdays and intra-division

Integer Programming

A mathematical optimization or feasibility program where some or all variables are restricted to integers.Let binary variables xijk = 1 if i plays j in slot k and xijk = 0 otherwise for i < j ∈ {1, …, n } and k ∈ { 1,… n – 1} for even n. In Basic SRRTP the IP problem should satisfy the following constraints :The first constraint guarantees that every team plays exactly once in each slot. The second ensures that each team plays every opponent exactly once.

Constraints Programming

Constraints Programming is a programming paradigm with a combinatorial approach for solving hard optimization problems.

CP for sports scheduling, as implemented by Martin

Henz

, performed the steps differently from IP. It first assigned teams to patterns before matching teams in games. This constraints programming solution for sports programming reported ran on the order of minutes as compared to the 24 hour run time from the Integer

Programming

Traveling Tournament Problem

Input

: A set of teams T = { 1, …,

n

}; D: an

n x n

integer distance matrix with elements

d

ij

;

l

,

u

integer parameters.

Output: A double round robin tournament on the teams in

T

such that

The length of each home stand and road trip is between

l

and

u

inclusive

The total distance traveled by the teams is minimized

With the maximum value of

u

= (n – 1) , this problem becomes a traveling salesman problem. For a small

u

the team must return home often, increasing travel distance. For

u

= 1, the problem becomes on of finding a feasible solution.

Combining IP and CP to Solve TTP

To solve this problem, a hybrid method using both Integer Programming and Constraints Programming was used. These two different models are used in a parallel algorithm known as the parallel branch and price algorithm. Each individual team tour is represented as the columns of this column generating algorithm. The Constraints Programming is used o solve the pricing problem to determine the solution to the minimum price, in this case, travel distance. The Integer Processor will attempt to determine the tours for each team in the tournament that will best solve the home stand/road trip constraint of the problem. Additionally a CP model is also used as part of a primal heuristic that is run on a separate processor on the side that will attempt to look for a good, but not necessarily best solution by using the solutions generated so far by the other parts of the algorithm.

Metaheuristics: Simulated Annealing

Simulated Annealing is a generic probabilistic metaheuristic for global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. The algorithm starts from a current state S. It probabilistically decides between moving from the current state to some neighboring S’ state. In this TTSA algorithm, the regions containing infeasible schedules penalize the algorithm to discourage it from remaining in the area.