Applied Mathematics IA

Name: Anthony AlexanderTeacher: Mr. David AkallooClass: 61 MathSubject: Applied Mathematics

Introduction

Football, or soccer, is a sport played between two teams of eleven players with a spherical ball. The game is played on a rectangular field of grass or green artificial turf, with a goal in the middle of each of the short ends. The object of the game is to score by driving the ball into the opposing goal. In general play, the goalkeepers are the only players allowed to touch the ball with their hands or arms, while the field players typically use their feet to kick the ball into position, occasionally using their torso or head to intercept a ball in midair. The team that scores the most goals by the end of the match wins.

Statement of Task

The annual Sports Day of Presentation College, San Fernando is soon approaching and one of the events this year is football. In the matches to be played, the teams must consist of eight (8) players: A goalkeeper, two (2) defenders, three (3) midfielders and two (2) strikers. The purpose of this study is to determine the best eight-man squad possible using the Hungarian Algorithm. The Hungarian Algorithm is used to assign jobs one-by-one to a specific person and to identify the best solution possible. Each person must have only one job and it is assumed each person is capable of handling the job.

Data Collection

Each player was placed in one of the various positions and was observed carefully. This was done in order to judge their abilities and determine their strengths. After this analysis, the players were rated on a scale of 1-10 and then chosen for the position which they fit best. The positions are in this order: Goalkeeper (GK), Left Back (LB), Right Back (RB), Left Wing (LW), Centre Attacking Midfielder (CAM), Right Wing (RW), Left Striker (LS) and Right Striker (RS). The players will be referred to as A, B, C, D, E, F, G and H.

Table showing the positions and the ratings of each playerGKLBRBLWCAMRWLSRS

A572610178

B712480109

C16577349

D810749873

E25127162

F69992203

G46367854

H16068588

Analysis of Data

The Hungarian AlgorithmAssumption: There are n players and n jobs. Step 0: If necessary, convert the problem from a maximum assignment into a minimum assignment. This is done by letting C = maximum value in the assignment matrix. Subtract all the values in the matrix from C. Step 1: From each row subtract off the row minimum. Step 2: From each column subtract off the row column minimum. Step 3: Use as few lines as possible to cover all the zeroes in the matrix. There is no easy rule to do this basically trial and error.Suppose you use k lines If k < n, let m be the minimum uncovered number. Subtract m from every uncovered number. Add m to every number covered with two lines. Go back to the start of step 3 If k = n, go to step 4 Step 4: Starting with the top row, work your way downwards as you make assignments. An assignment can be (uniquely) made when there is exactly one zero in a row. Once an assignment is made, delete that row and column from the matrix. If you cannot make all n assignments and all the remaining rows contain more than one zero, switch to columns. Starting with the left column, work your way rightwards as you make assignments. Iterate between row assignments and column assignments until youve made as many unique assignments as possible. If you still havent made n assignments and you cannot make a unique assignment either with rows or columns, make one arbitrarily by selecting a cell with a zero in it. Then try to make unique row and/or column assignments.GKLBRBLWCAMRWLSRSC

A57261017810

B71248010910

C1657734910

D81074987310

E2512716210

F6999220310

G4636785410

H1606858810

Table showing values and the maximum value C

GKLBRBLWCAMRWLSRSRow Min.

A538409320

B3986210010

C945337611

D203612370

E859839483

F4111881071

G647432562

H9410425222

Table showing the row minimum and the values after subtracting the values from C

GKLBRBLWCAMRWLSRS

A53840932

B398621001

C83422650

D20361237

E52650615

F30007796

G42521034

H72820300

Col.Min.20000000

Table showing the column minimum and the values after subtracting the row minimum

GKLBRBLWCAMRWLSRS

A33840932

B198621001

C63422650

D00361237

E32650615

F10007796

G22521034

H52820300

Table showing the values after subtracting the column minimum

GKLBRBLWCAMRWLSRS

A33840932

B198621001

C63422650

D00361237

E32650615

F10007796

G22521034

H52820300

Table showing the zeroes being covered by red lines and the minimum uncovered number covered by a blue line

GKLBRBLWCAMRWLSRS

A22730821

B198631001

C63423650

D00362237

E21540504

F10008796

G22522034

H52821300

Table showing m subtracted from uncovered numbers and added to numbers covered by two lines.

GKLBRBLWCAMRWLSRS

A11620720

B19863900

C63424660

D00363247

E10430403

F100097106

G22523044

H52822310

GKLBRBLWCAMRWLSRS

A11400720

B19643900

C63204660

D00143247

E10210403

F3200119128

G22303044

H52602310

GKLBRBLWCAMRWLSRS

A11400720

B19643900

C63204660

D00143247

E10210403

F3200119128

G22303044

H52602310

D GK

LBRBLWCAMRWLSRS

A1400720

B9643900

C3204660

E0210403

F200119128

G2303044

H2602310

E LB

RBLWCAMRWLSRS

A400720

B643900

C204660

F00119128

G303044

H602310

F RB

LWCAMRWLSRS

A00720

B43900

C04660

G03044

H02310

A CAM

LWRWLSRS

B4900

C0660

G0044

H0310

G RW

LWLSRS

B400

C060

H010

B LS

LWRS

C00

H00

C RSH LW

Conclusion

After implementing the Hungarian Algorithm, the most suitable player was determined for each position. The players and their positions are: A CAM B LS C RS D GK E LB F RB G RW H LW In the end, both C and H were suitable for both positions. Therefore they were chosen for their position arbitrarily. The purpose of this study was achieved and the best team possible was determined.