This article is all about study and research on sports and sports management. The article is written by group of young authors.

1. NMIMS
Anil Surendra Modi School of Commerce
A Study on the application of Operation
Research in Sports and Sports Management
SY BSc. Finance B
Prof. Tehreem Bardi
4th
October 2017
Group 2
Ansh Chawla –B006 (Sap Id-74051016045)
Anusha Gupta-B007 (Sap Id- 74051016075)
Anushka Gupta-B008 (Sap Id- 74051016076)
Arundhati kele-B009 (Sap Id-74051016112)
Aryaman Kothari-B010 (Sap Id-74051016121)
Sign:
Marks:

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Acknowledgement
We would like to express our gratitude to Professor Tehreem Bardi for giving us this
opportunity to make this report and for guiding us and clearing our doubts during her
lunch break.
Researching for this project has been an eye opening experience as we have realised
that we have not even scratched the surface when it comes to the kind of research that
need to be done while publishing a research paper or article. This report has helped us
understand the road that lies ahead of us and put us on the correct path towards
achieving it.
Executive Summary
This Essay will talk about the various applications of Operations Research in sports
and sports management and why research in this filed is so relevant today. This field
is an extremely profitable and interesting field that has attracted various researchers to
conduct studies to optimise tactics and strategies and also to optimise schedules for
the players and officials (referees, Umpires). This will be seen with the help of
examples and studies conducted by Laura Albert McLay, F. Della Croce, R. Tadei
and P.S. Asioli and Mario Guajardo and Denis Saure .The essay will also talk
about using OR for forecasting future outcomes in sports by highlighting the various
key researches conducted in the past decades. The essay will be concluded by
reflecting on the things discussed and also by looking towards the future of
Operations Research in sports.

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INDEX
SERIAL
NO. TOPICS
PAGE
NO.
1. Introduction 4
2. Tactics and Strategy 4
3. Scheduling 8
4. Forecasting 11
5. Conclusion 13
6. Bibliography 14
7. Appendix 15

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INTRODUCTION
Operations research is the use of analytical techniques and models to improve
decision making in various fields such as business, logistics, marketing and even
sports.1
This Essay will talk about the use of Operations research in sports and sports
management and why it is relevant. Sports competitions involve many logistical and
economic issues such as tournament planning, club management, marketing policies,
security issues etc. Today sports is a big business, Forbes valued the Major League
Baseball teams at over $15 billion, with the Yankees alone worth $1.7 billion. With
such values it is not surprising that there is a great interest in using data and OR
models to make better decisions. Lots of sports leagues around the world also have
high economic effects, making the overall sports economy a significant part of the
overall economy.2
This essay will examine few areas of study such as analysis of
tactics and strategy, scheduling and forecasting.
Tactics and strategy – Over the years various people have researched ways to
optimize the strategies for different sports. Some examples include Freeze (1974) who
used a simulation to find the best batting order for a baseball team, Boon and
Sierksma (2003), they produced a decision support system to help football coaches
and managers assess contributions of particular players to their teams, hence helping
not only in team selection but also in scouting and purchasing new players, Scarf and
Grehan, (2005) determined the optimal route to choose while cycling.3
1 University, Lancaster. “What is Operational Research? | Lancaster University.” Lancaster University
Management School, www.lancaster.ac.uk/lums/study/masters/programmes/msc-operational-research-
management-science/what-is-operational-research/.
2 “The Appeal of Operations Research and Sports.” Michael Trick's Operations Research Blog, 6 Nov.
2013, mat.tepper.cmu.edu/blog/?p=1409.
3 50 years of OR in sports. eprints.lancs.ac.uk/45286/1/10.pdf.

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A tactical example of OR is the decision of weather a team should run or pass in a
game of American football, This can be solved using Linear programming.
Linear programming (also called linear optimization) is a method of choosing the best
alternative from a set of feasible alternatives (such as maximum profit or lowest cost)
in situations in which the objective function as well as the constraints can be
expressed as linear mathematical functions. In order to apply linear programming
certain requirements need to be met.
There should be an objective that should be clearly identifiable and measurable in
quantitative terms. It could be maximization of sales/ profit, minimization of cost etc.
The activities to be included should be distinctly identifiable and measurable in
quantitative terms.
The resources that are to be allocated for the fulfillment of the goal should also be in
identifiable and measurable quantitatively. They must be in limited supply. The
technique would involve allocation of these resources in a manner that would trade
off the returns on the investment of the resources for the attainment of the objective.
There should be a series of feasible alternative courses of action available to the
decision makers, which are determined by the resource constraints.
Some assumptions of Linear Programming:
1. Divisibility - Decision variables can be fractions. The values only make sense
when they are integers; after that we need an extension of linear programming,
which is known as integer programming.
2. Certainty - The model assumes that the responses represented by the
coefficients are equal to the values of the variables.
3. Data - To formulate a linear program it is assumed that the data is available to
specify the problem.
4. The objective function and constraints are linear.

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So coming back to the problem “Should a Football team run or pass?” Laura Albert
McLay conducted a research to solve this.
Problem: The offense can run or pass the ball, The defense can anticipate the choice
of the offense and choose whether to run or pass offense, Given this information what
is the best mix of pass and run plays for the offense and defense.
In a two-player example a Payoff Matrix is created between row player R and column
player C. It is defined by m*n payoff matrix having a sum of zero.
A= [ 0 1 -1 ]
[-1 0 1 ]
[ 1 -1 0 ]
Example: Offense vs Defense (yards)
A(x,y) Offense runs(x) Offense passes(y)
Run Defense(v) -5 10
Pass Defense(w) 5 0
y = 1 - x
w= 1 - v
The offense wants the most yards .The defense wants the offense to have the fewest
yards. The sum of this payoff matrix is Zero hence it is a zero-sum game.
Case 1:Offense
Max Z
Subject to equations:
Z<= -5x + 10y (1)
Z<= 5x (2)

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x + y = 1
x , y >= 0
Substituting y in equation 1:
Z<= 10 – 15x
We obtain the largest value of Z when x= ½ and y = ½ ,
Hence run half the time and pass half the time using graphical representation.
Therefore Z= 2.5 yards per play.
Case 2:Defence
Min U
Subject to equations:
U<= 5 – 10v
U>= 10v
v + w = 1
v , w >= 0
We want the smallest value of U and this occurs when
v = ¼ and w = ¾ , Hence run a quarter of the time and pass
Three quarters of the time
Therefore U = 2.5 yards per play
Conclusion: The offense gain and defense loss are always identical.4
Scheduling – Scheduling competitions is another important application of OR in
sports as Sports teams need schedules that satisfy different types of constraints.
4 “Should a football team run or pass? A game theory and linear programming approach.” Punk
Rock Operations Research, 8 Oct. 2015, punkrockor.com/2015/10/08/should-a-football-team-
run-or-pass-a-game-theory-and-linear-programming-approach/.

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Optimization of such a Timetable is done using several Operations research
techniques.5
In practise while making a schedule there are many objectives and constraints thus
specialized software’s are often required, using an OR algorithm of some kind.
Constraints are derived from issues like ground availability and the requirements of
TV companies or sponsors. Objectives can include minimizing distance travelled,
producing home/away patterns, minimizing dates when two particular teams are both
at home, etc. Additionally there may be several types of preference expressed by the
teams that need to be assigned relative weights. One method to find an optimized
schedule is using integer programming.
An integer-programming problem is a mathematical optimisation or practicability
program during which some or all of the variables are restricted to be integers (cant be
in fractions like linear programming). This method is used when fractional solutions
are unrealistic. It has an objective function (maximization or minimization) and
constraints that are linear.
There are two types:
1. Mixed integer linear programming (MILP)- It involves issues in which just
some of the variables are strained to be integers, whereas different variables are
allowed to be non-integers. The integer variables represent quantities that may solely
be integer.
2. Zero-one linear programming or Binary programming - It involves issues in
which the variables are restricted to be either zero or one. Note that any finite integer
variable is expressed as a mix of binary variables. The integer variables represent
choices that ought to solely take on the value zero or one.
5 Operations Research Management Science - OR in Sports, www.orms-today.org/orms-6-
04/sports.html.

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For example the management ahs to determine whether or not to engage in the
following activities: (i) to build a new plant, (ii) to undertake an advertising
campaign, or (iii) to develop a new product. In each case, we must make a yes–no or a
go–no–go decision. These choices are easily modeled by letting X j =1 if we engage
in the jth
activity and X j =0 if we don’t. Variables that are restricted to 0 or 1 in this
way are termed binary, bivalent, logical, or 0–1 variables. Binary variables are of
great importance because they occur regularly in many model formulations,
particularly in problems addressing long-range and high-cost strategic decisions
associated with capital-investment planning.
Many other solution approaches have been proposed to solve the scheduling
constraints with varying degrees of success: integer linear programming, constraint
programming, local search (simulated annealing, hybrid approach).6
A study by F. Della Croce, R. Tadei and P.S. Asioli looks into the use of integer
programming to optimize a round robin tennis tournament schedule.
A round robin tournament is when all teams play all the teams. In case of Tennis each
participant plays every other participant once. If they play each participant twice then
it’s called a double round robin.
A practical problem encountered by the management of a tennis club is the
organization of a tennis tournament for the club members. The tournament
participants are divided into different series: in each series, every player plays once a
week with a different opponent in a round robin tournament. All matches are subject
to a time limit of one hour. The series share the same set of courts, whose weekly
availability is pre-decided. In addition, the players have their own availability
constraints. Given the courts and players availability, the objective is to schedule the
6 “Integer programming.” Wikipedia, Wikimedia Foundation, 25 Sept. 2017,
en.wikipedia.org/wiki/Integer_programming.

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tournament with no violation of the constraints or, more realistically, in order to
maximize the number of feasible matches. This problem can be formulated as a
maximum matching problem, with the additional constraint that each player must play
just once a week. A two-step procedure is used; first, the round robin tournaments of
each series are generated then the matches of each tournament are assigned to the
available courts for every week by means of a local search procedure.7
Similarly a study was conducted by Mario Guajardo and Denis Saure using Integer
programming to determine a schedule for football tournaments with constraints for
Home/Away, weather, stadium availability, tourism related and TV related
constraints. This study was able to incorporate multiple objective functions like to
maximize the concentration of games between teams in the same group toward the
final rounds of the tournament, Maximizing the number of good trips, minimizing the
distance travelled by the teams (Travelling tournament problem).
An example of a constraint formed in integer programming can be seen for when
Each team plays each of the others once over the course of the tournament.
𝑥𝑖𝑗𝑘 + 𝑥𝑗𝑖𝑘 = 1
!∈!
, ∀𝑖, 𝑗 ∈ 𝐼, 𝑖 ≠ 𝑗
Decision variables
𝑥𝑖𝑗k = 1 (If team i plays at home against team j in round k) else 𝑥𝑖𝑗k = 0
𝑖, 𝑗 ∈ 𝐼: 𝑠𝑒𝑡 𝑜𝑓 𝑡𝑒𝑎𝑚 : {Team 1, Team 2, Team 3, . . .}
𝑘 ∈ 𝐾: 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑜𝑢𝑛𝑑 : {Round 1,Round 2,Round 3...}
7
https://www.researchgate.net/publication/242916472_Scheduling_a_round_robin_tennis_tournamentu
nder_courts_and_players_availability_constraints [accessed Oct 2, 2017].

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Integer programming can also be used for assigning referees to a schedule of matches
in order to satisfy a number of conditions e.g. Balancing the frequency of referees vs
teams, Balancing the total number of games assigned to each referee, Balancing the
average travel distances of the referees, Taking into account the experience of the
referees to officiate some particular games. Hence Integer programming can also be
used to solve this keeping in mind some pre-defined assignments and unavailable
referees per round.
Forecasting- It is the process of making predictions of the future based on past and
present data and most commonly by analysis of trends. OR has made many
contributions to forecasting research and practice. But the last 25 years have seen the
rapid growth of specialist forecasting research. The unique contribution that OR can
continue to make to forecasting is through developing models that link the
effectiveness of new forecasting methods to the organizational context in which the
models will be applied. In many real-life sports games, spectators are interested in
predicting the outcomes and watching the games to verify their predictions.
Traditional approaches include subjective prediction, objective prediction, and simple
statistical methods.
Gambling has always been a major feature of most sports. Because of the amounts of
money involved, it is likely that all sorts of OR modeling is taking place within
bookmakers’ organizations which is being regarded as highly confidential. There are
various researchers that contributed to this area and below given are some important
researchers and their contributions.
For many years Clarke (1993) used to predict the results of Australian Rules football
matches, using the Hooke and Jeeves method. More recently, Flitman (2006) used
neural nets and Linear Programming for the same purpose. Dixon and Robinson
(1998) reported on a statistical model used to predict the results of football matches
and to update those predictions during the course of a match. Klaassen and Magnus
(2003) and Barnett and Clarke (2005) used statistical analysis to predict the winner of

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a tennis match. Other work includes Philpott et al (2004), who used simulation to
forecast the winners of yachting races, and Lo et al (1995), who used ranking
probability models to help those who gamble on horse races. Aside from gambling,
two researches described methods for predicting success at the Olympic Games:
Condon et al (1999), who used regression and neural nets to forecast national success,
and Heazlewood (2006), who reported on non-linear models used to predict winning
times and distances for athletics and swimming. This application could be very
important for countries looking to increase their success arte at the Olympics.
Another researcher Adler et al. developed integer-programming models to calculate
the number of games a team has to win to reach a playoff spot. These models can
determine earlier than the method used by the press when a particular team is to be
eliminated.
They developed two integer-programming models that can detect in advance when a
team is already qualified for, or eliminated from, the playoffs. These models are based
on the computation of the guaranteed qualification score (GQS), which is the
minimum number of points a team has to obtain to be sure it will be qualified,
regardless of any other results, and the possible qualification score (PQS), which
consists of the minimum number of points a team has to win to have any chance to be
qualified. A team is mathematically qualified for the playoffs if and only if its number
of points won is greater than or equal to its GQS. Only at this point can its
qualification for the playoffs be announced without any risk of contradiction. A team
is mathematically eliminated from the playoffs if and only if its current number of
points plus the number of points it is still able to win in the remaining games to be
played is smaller than its PQS. An additional feature of these models is that they can
be easily altered to accommodate some of the usual tie-breaking rules.
These models were applied to the 2002 edition of the Brazilian national soccer
championship played by the 26 major teams from Aug. 10 to Dec. 15.

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CONCLUSION
Operations research in sports is a vast field that has been researched over the past few
decades in areas such as tactics and strategies, scheduling and forecasting. However
this is not the end as there are other fields of study like the unorganized sports area.
There will also be Sophisticated OR software that will make further inroads into the
scheduling of fixtures and officials (and possibly other things such as training
facilities) not only for professionals but also at amateur level as well. There is now a
huge amount of data available for a growing number of sports, capturing almost every
facet of a sports encounter on a computer in a form susceptible to quantitative
analysis. This enables sophisticated OR models to be of practical value to decision-
makers within sport. This trend will continue, and it may well be that OR will become
an accepted tool in helping players, coaches and others to plan and play their games.

15. 15
Bibliography
1) University, Lancaster. “What is Operational Research? | Lancaster
University.” Lancaster University Management School,
www.lancaster.ac.uk/lums/study/masters/programmes/msc-operational-
research-management-science/what-is-operational-research/.
2) “The Appeal of Operations Research and Sports.” Michael Trick's Operations
Research Blog, 6 Nov. 2013, mat.tepper.cmu.edu/blog/?p=1409.
3) 50 years of OR in sports. eprints.lancs.ac.uk/45286/1/10.pdf.
4) “Should a football team run or pass? A game theory and linear
programming approach.” Punk Rock Operations Research, 8 Oct. 2015,
punkrockor.com/2015/10/08/should-a-football-team-run-or-pass-a-
game-theory-and-linear-programming-approach/.
5) Operations Research Management Science - OR in Sports, www.orms-
today.org/orms-6-04/sports.html.
6) “Integer programming.” Wikipedia, Wikimedia Foundation, 25 Sept. 2017,
en.wikipedia.org/wiki/Integer_programming.
7) https://www.researchgate.net/publication/242916472_Scheduling_a_round_ro
bin_tennis_tournamentunder_courts_and_players_availability_constraints
[accessed Oct 2, 2017].

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Appendix 1

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Appendix 2

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Appendix 3

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