Applied Mathematics IA
Name: Anthony Alexander Teacher: Mr. David Akalloo Class: 61 Math Subject: 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 player A B C D E F G H
GK 5 7 1 8 2 6 4 1
LB 7 1 6 10 5 9 6 6
RB 2 2 5 7 1 9 3 0
LW 6 4 7 4 2 9 6 6
CAM 10 8 7 9 7 2 7 8
RW 1 0 3 8 1 2 8 5
LS 7 10 4 7 6 0 5 8
RS 8 9 9 3 2 3 4 8
Analysis of Data The Hungarian Algorithm Assumption: 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 you’ve made as many unique assignments as possible. If you still haven’t 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. GK LB RB A 5 7 2 B 7 1 2 C 1 6 5 D 8 10 7 E 2 5 1 F 6 9 9 G 4 6 3 H 1 6 0 Table showing values and
GK
LB
RB
LW CAM 6 10 4 8 7 7 4 9 2 7 9 2 6 7 6 8 the maximum
LW
A 5 3 8 4 B 3 9 8 6 C 9 4 5 3 D 2 0 3 6 E 8 5 9 8 F 4 1 1 1 G 6 4 7 4 H 9 4 10 4 Table showing the row minimum the values from C
CAM 0 2 3 1 3 8 3 2 and
RW 1 0 3 8 1 2 8 5 value
RW
LS 7 10 4 7 6 0 5 8 C
RS 8 9 9 3 2 3 4 8
LS
RS
C 10 10 10 10 10 10 10 10
Row Min. 9 3 2 0 10 0 1 0 7 6 1 1 2 3 7 0 9 4 8 3 8 10 7 1 2 5 6 2 5 2 2 2 the values after subtracting
GK 5 3 8 2 5 3 4 7 2
LB 3 9 3 0 2 0 2 2 0
RB 8 8 4 3 6 0 5 8 0
LW 4 6 2 6 5 0 2 2 0
CAM 0 2 2 1 0 7 1 0 0
RW 9 10 6 2 6 7 0 3 0
LS 3 0 5 3 1 9 3 0 0
A B C D E F G H Col.Mi n. Table showing the column minimum and the values after subtracting the row minimum
GK A 3 B 1 C 6 D 0 E 3 F 1 G 2 H 5 Table showing
A B C D E F G H
GK 3 1 6 0 3 1 2 5
RS 2 1 0 7 5 6 4 0 0
LB RB LW CAM RW LS RS 3 8 4 0 9 3 2 9 8 6 2 10 0 1 3 4 2 2 6 5 0 0 3 6 1 2 3 7 2 6 5 0 6 1 5 0 0 0 7 7 9 6 2 5 2 1 0 3 4 2 8 2 0 3 0 0 the values after subtracting the column minimum
LB 3 9 3 0 2 0 2 2
RB 8 8 4 3 6 0 5 8
LW 4 6 2 6 5 0 2 2
CAM 0 2 2 1 0 7 1 0
RW 9 10 6 2 6 7 0 3
LS 3 0 5 3 1 9 3 0
RS 2 1 0 7 5 6 4 0
Table showing the zeroes being covered by red lines and the minimum uncovered number covered by a blue line
GK LB RB LW CAM RW LS A 2 2 7 3 0 8 2 B 1 9 8 6 3 10 0 C 6 3 4 2 3 6 5 D 0 0 3 6 2 2 3 E 2 1 5 4 0 5 0 F 1 0 0 0 8 7 9 G 2 2 5 2 2 0 3 H 5 2 8 2 1 3 0 Table showing ‘m’ subtracted from uncovered numbers and to numbers covered by two lines.
A B C D E F G H
A B C D E F G H
RS 1 1 0 7 4 6 4 0 added
GK 1 1 6 0 1 1 2 5
LB 1 9 3 0 0 0 2 2
RB 6 8 4 3 4 0 5 8
LW 2 6 2 6 3 0 2 2
CAM 0 3 4 3 0 9 3 2
RW 7 9 6 2 4 7 0 3
LS 2 0 6 4 0 10 4 1
RS 0 0 0 7 3 6 4 0
GK 1 1 6 0 1 3 2 5 GK
LB 1 9 3 0 0 2 2 2 LB
RB 4 6 2 1 2 0 3 6 RB
LW 0 4 0 4 1 0 0 0 LW
CAM 0 3 4 3 0 11 3 2 CAM
RW 7 9 6 2 4 9 0 3 RW
LS 2 0 6 4 0 12 4 1 LS
RS 0 0 0 7 3 8 4 0 RS
A 1 B 1 C 6 D 0 E 1 F 3 G 2 H 5 D GK
A B C E F G H E LB
A B C F G H F RB
A B C G H
1 9 3 0 0 2 2 2
4 6 2 1 2 0 3 6
0 4 0 4 1 0 0 0
0 3 4 3 0 11 3 2
7 9 6 2 4 9 0 3
2 0 6 4 0 12 4 1
LB 1 9 3 0 2 2 2
RB 4 6 2 2 0 3 6
LW 0 4 0 1 0 0 0
CAM 0 3 4 0 11 3 2
RW 7 9 6 4 9 0 3
LS 2 0 6 0 12 4 1
RS 0 0 0 3 8 4 0
RB 4 6 2 0 3 6
LW 0 4 0 0 0 0
CAM 0 3 4 11 3 2
RW 7 9 6 9 0 3
LS 2 0 6 12 4 1
RS 0 0 0 8 4 0
LW 0 4 0 0 0
CAM 0 3 4 3 2
RW 7 9 6 0 3
LS 2 0 6 4 1
RS 0 0 0 4 0
A CAM
0 0 0 7 3 8 4 0
B C G H
LW 4 0 0 0
RW 9 6 0 3
LS 0 6 4 1
LW 4 0 0
LS 0 6 1
RS 0 0 0
LW 0 0
RS 0 0
RS 0 0 4 0
G RW
B C H B LS
C H C RS
H 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.
