ACTIVITY 1
WACE Exam: Question 3 (calculator-free) - Hungarian algorithm
A foreman in a factory has four workers, Adam, Ben, Cate and Demi, and four jobs to
complete. The time, in hours each worker can complete a particular job is given in the
weighted bipartite graph below.
(a)
Complete the matrix associated with the bipartite graph above.
J1 J2
A
B
C
D
(b)
4


3


6
7
J3
J4
7
3



2

6
Using the Hungarian algorithm, determine which job the foreman should assign to
each of his workers so that the total time is minimised.
J1 J2
A 4
B  2
C 3

D 9
J3
J4
7
7
6
3
4
5
7
4
8
4 
2

6
ACTIONS

From each number in each row, subtract the smallest number in that row

Now within each column subtract the smallest number in the column from every
number in the column

(only column 2, other minimum numbers are 0)
** Using vertical &/or horizontal lines, cross out rows and columns so that all the
zeroes are crossed out using as few lines as possible
(can be done with 3 lines)

Identify the smallest number that is not crossed out
(1)

To create the next matrix, take this minimum number away from all the uncovered
numbers and add it to the numbers at the intersections of the vertical and horizontal
lines. Other covered numbers are unchanged.

Repeat **

As 4 lines are needed to cross out all the zeroes and the matrix is a 4 x 4 matrix, then
a solution exists

To assign people to jobs, start by locating rows or columns with only one zero –
these are the only jobs for these people (or people for this job)

Note: Row 4, column 3 and column 4, row 3. Assign these and take them out of
circulation. Allocate the last jobs. Check the question has been answered.
ACTIVITY 2: For beginners
Four teachers: Bart, Lisa, Homer and Maggie each do one task - write the test, mark the
test, prepare the solutions and write the report. They each take different times for the various
tasks and these times are shown in the table provided. How can you assign the tasks to
minimise the total time taken?
Write test
Mark test
Prepare solutions
Write report
Bart
8
5
4
13
Lisa
10
6
7
10
Maggie
8
9
9
11
Homer
11
7
3
6
Write test
Mark test
Prepare solutions
Write report
Bart
Lisa
Maggie
Homer







From each number in each row, subtract the smallest number in that row
Within each column subtract the smallest number in the column from every number in the column
Using vertical &/or horizontal lines, cross out rows and columns so that all the zeroes are crossed
out using as few lines as possible
As 4 lines are needed to cross out all the zeroes and it is a 4 x 4 matrix, then a solution exists
To assign people to jobs, start by locating rows or columns with only one zero – these are the
only jobs for these people (or people for this job)
As tasks are assigned to people, remove them from consideration.
Allocate all tasks. Check the question has been answered.
ACTIVITY 3: Multiple steps and more than one solution
A catering company has four Managers (M1 to M4) based in four different branches. These
Managers are to be allocated to four different Events designated as A, B, C and D. The
distances in kilometres from each branch to each Event are given in the table. Use the
Hungarian algorithm to determine how the Managers should be allocated to these Events so
that the total distance travelled by them is minimised.
Event
A
B
C
D
M1
40
25
25
30
M2
10
60
30
40
M3
45
15
10
25
M4
15
80
65
85
Event
A
B
C
D
Event
A
B
C
D
Manager
Manager
M1
M2
M3
M4
Manager
M1
M2
M3
M4
ACTIONS FOR ACTIVITY 3

From each number in each row, subtract the smallest number in that row

Now within each column subtract the smallest number in the column from every
number in the column

(only column 4, other minimum numbers are 0)
** Using vertical &/or horizontal lines, cross out rows and columns so that all the
zeroes are crossed out using as few lines as possible

(can be done with 3 lines)
If the number of lines is equal to the dimensions of the matrix the solution exists – Go
to the ASSIGNMENT.

If the number of lines is less than the dimensions of the matrix the solution is yet to
be “found”.

Identify the smallest number that is not crossed out

Create the next matrix. Take this minimum number away from all the uncovered
numbers and add it to the numbers at the intersections of the vertical and horizontal
lines. Enter other covered numbers unchanged.

Go back to the Step **

ASSIGNMENT - To assign managers to events, start by locating rows or columns
with only one zero – these are the only events for these managers (or managers for
these events)

Note: For this assignment problem, there is more than one solution.
o
How do you know there is more than one solution?
o
List all possible solutions.
o
State the minimum total distance for the Managers to get to the events.
o
Check the question has been answered.
Activity 4: Maximising – Outings for the elderly
An optimum solution may involve assignment so that a value is maximised rather than
minimised. Start by creating a new matrix in which every number in the original matrix
is subtracted from the largest number in that matrix. Then proceed as for a minimum.
The local council wants as many elderly people as possible to make use of the services they
offer for outings. The outings involve going shopping, taking people out to lunch, taking
people bowling and going for a drive to the country. The results of their survey of people’s
preferences for the four different activities on the days available are recorded in the table
below. To maximise the number of people using their service, on which days should each of
the activities be scheduled?
Monday
Tuesday
Wednesday
Thursday
Shopping
17
24
42
21
Lunch
25
18
19
20
Bowling
29
14
31
22
Driving
11
20
17
14
Monday
Tuesday
Wednesday
Thursday
Shopping
Lunch
Bowling
Driving
Activity 4: Maximising – An engineering competition
An optimum solution may involve assignment so that a value is maximised rather than
minimised. Start by creating a new matrix in which every number in the original matrix
is subtracted from the largest number in that matrix. Then proceed as for a minimum.
Four students are to be chosen for four different tasks in an engineering competition.
To select the students for each task, the four applicants were given a sample task
which was marked by the teacher. The marks are recorded in the table below. Which
is the best allocation of students to tasks so that the team performance is
maximised?
Build bridge Design tower Plan project Scale model
Tim
60
78
67
37
Alf
45
80
70
90
Sara
60
35
70
86
Jess
42
66
54
72
Build bridge Design tower Plan project Scale model
Tim
Alf
Sara
Jess
ACTIVITY 5: More tasks than people – Cleaning up the Art room






There are more people than tasks
Number of supply sources exceeds number of demand depots
The matrix formed is not square
We make it square by adding a row or column
By convention the row or column added is filled with the largest number in the matrix
In Sadler the extra row or column is filled with 0s
Five students have been cleaning up the Art room after class and their times taken to do
various tasks have been recorded. How can these five students be allocated to the tasks so
that the total time taken to complete this part of the cleanup is minimised?
Pack away the
paint
Gather up
paper and
card
Wipe the
benches
Sweep the
floor
Toni
10
19
8
15
Eva
10
18
7
17
Max
13
16
9
14
Carol
12
19
8
18
Ned
14
17
10
19
Pack away the
paint
Gather up
paper and
card
Wipe the
benches
Sweep the
floor
Toni
Eva
Max
Carol
Ned
ACTIVITY 6: A 5 x 5 matrix
Five students form a team which is entered into a problem-solving competition
between schools. From the local Algebra School five students have recorded their
times to solve the types of problems and puzzles used in the competition. Which is
the best allocation of students to the types of problems so as to minimise the total
time needed for the team to complete these tasks consecutively?
Logic
Sudoku
Ken-ken
Rubics
Battleships
Nick
1
3
4
5
7
Ava
8
3
12
13
5
Sam
11
12
4
8
5
Lee
6
13
12
5
5
Vo
14
13
16
10
10
Logic
Sudoku
Ken-ken
Rubics
Battleships
Nick
Ava
Sam
Lee
Vo
ACTIVITY 7: Investigating the Hungarian Algorithm
How can you manipulate the numbers in a matrix to create a new set of numbers to use for
the Hungarian Algorithm? Does the new set of numbers give the same assignment solution
as the original matrix?
Example 1
Teachers can go to one of three locations for an afterschool PD. The table below shows the
distance from the PD to their homes. By inspection, minimise the total distance travelled by
the three teachers. Justify your choice.
Teachers
Morley
Gwelup
Warwick
Ali
10
4
9
Joy
8
11
10
Ron
9
8
7
Example 2
Gabriel has three contractors (C1, C2, C3) who work from home. Three jobs (J1, J2 and J3)
need to be assigned and the time needed for each contractor to travel to each job is
summarised in the table. Determine the optimum assignment to minimise the total travel
time.
Contractors
J1
J2
J3
C1
20
15
30
C2
30
15
20
C3
30
25
40
For each of these matrices determine the effect on the assignment solution for each of the
following manipulations.
1.
Add the same number to every number in the matrix
2.
Add a different number to the elements in each row
3.
Multiply all the numbers in the matrix by the same number
4.
Multiply every number in each row by a different number for each row.
Share your findings.