Assignment Problem
Hungarian Method.
Example : Solve the following assignment problem
1
2
3
4
A
8
12
12
14
B
2
12
14
6
C
4
10
8
4
D
6
4
2
8
Solution
1. Subtract the smallest element from each row, then subtract the smallest element
from each column.
2. Cover all zeros with the least possible number of lines. If the number of lines is
equal to the number of rows, you are optimal.
3. If not, look at the smallest number not covered by a line. Add this to every number
covered by two lines; subtract it from every number not covered by any line. Repeat
steps 2 and 3.
1
2
3
4
A
8
12
12
14
B
2
12
14
6
C
4
10
8
4
D
6
4
2
8
1
1
2
3
4
A
8
12
12
14
8
B
2
12
14
6
2
C
4
10
8
4
4
D
6
4
2
8
2
1
2
3
4
A
0
4
4
6
8
B
0
10
12
4
2
C
0
6
4
0
4
D
4
2
0
6
2
1
2
3
4
A
0
4
4
6
B
0
10
12
4
C
0
6
4
0
D
4
2
0
6
0
2
0
0
1
2
3
4
2
A
0
2
4
6
B
0
8
12
4
C
0
4
4
0
D
4
0
0
6
0
2
0
0
1
2
3
4
A
0
0
2
6
B
0
6
10
4
C
0
2
2
0
D
6
0
0
8
0
2
0
0
1
2
3
4
A
0
0
2
6
B
0
6
10
C
0
2
2
0
D
6
0
0
8
0
2
0
0
Total Minimum Cost = 12+2+4+2=20
Maximization Problem
3
To sign an assignment problem whose objective is to maximise, the problem should
first be converted to a minimisation problem before the application of the Hungarian
Method.
1. Convert the problem to a minimisation problem by multiplying all the elements
cij of the assignment matrix by-1.
2. If some of the elements of the cost matrix are negative ,add a sufficiently
large positive number to the corresponding rows and columns so that all the cost
elements would become nonnegative.
3. We now have an assignment problem with a minimisation objective and all cost
elements nonnegative. The Hungarian Method can now be applied directly
Solve the following maximisation assignment problem:
A
B
C
D
1
20
60
50
55
2
60
30
80
75
3
80
100
90
80
4
65
80
75
70
A
B
C
D
1
-20
-60
50
-55
2
-60
-30
80
75
3
-80
-100
-90
-80
4
-65
-80
-75
-70
4
A
B
C
D
1
80
40
50
45
2
40
70
20
25
3
20
0
10
20
4
35
20
25
30
A
B
C
D
1
80
40
50
45
40
2
40
70
20
25
20
3
20
0
10
20
0
4
35
20
25
30
20
A
B
C
D
1
40
0
10
5
40
2
20
50
0
5
20
3
20
0
10
20
0
4
15
0
5
10
20
5
A
B
C
D
1
40
0
10
5
2
20
50
0
5
3
20
0
10
20
4
15
0
5
10
15
0
0
5
A
B
C
D
1
25
0
10
0
2
5
50
0
0
3
5
0
10
15
4
0
0
5
5
15
0
0
5
TOTAL = 55+80+65+100= 300
3B
4A
2C
1D
6
Tutorial 1 (Assignment)
1. Find the optimal assignment of four jobs and four machines when the
cost of assignment is given by the following table
J1
J2
J3
J4
M1
10
9
8
7
M2
3
4
5
6
M3
2
1
1
2
M4
4
3
5
6
Answer: total minimum cost 20
M1-J3, M2-J1, M3-J4 ,M4-J2 OR M1-J3, M2-J4, M3-J2, M4-J1
7
2. Solve the following assignment problems with the objectives as stated:
(a) Maximise
j
1
2
3
4
5
1
28
20
36
36
28
i
2
25
30
32
33
30
3
35
23
36
37
33
4
33
25
32
33
35
5
34
26
40
42
35
4
1
2
8
4
10
5
5
7
9
5
1
(b) Minimise
j
1
2
3
4
5
1
4
1
7
10
2
i
2
2
3
4
0
8
3
0
5
2
3
9
(a) Max = 178 ,(1,3),(2,2),(3,1),(4,5),(5,4)
(b) Min =5 ,(1,4),(2,1),(3,3),(4,2),(5,5)
3.
(a)
(b)
Sometimes the number of agents in an assignment problem is
not equal to the number of tasks. Describe briefly how one can
adapt such a problem so that the Hungarian method can be
used to optimise it
Tony's Cash and Carry has just leased a new store and is
attempting to determine where the various departments should
be located within the store. The store manager has four
locations that could accommodate one department each, the five
departments under consideration being a shoe, a toy, an autoparts, a hardware, and an electronics department. After a careful
study of the layout of the store, and based on his experience of
similar stores, the store manager has made estimates of the
expected profit (in hundreds of thousands of pounds sterling) for
each department in each location.
Location
Department
1
2
3
4
Shoe
10
6
10
8
Toy
15
18
5
11
Auto-parts
17
10
13
16
Hardware
14
12
13
10
Electronics
14
16
6
12
Use the Hungarian method to determine
(i)
a choice of departments and their locations that will
achieve the maximum possible expected profit;
(ii)
the maximum possible expected annual profit attainable,
(iii)
whether any alternative optimal solution exists.
8
9
4. A company is in the process of deciding which of five salespeople to
assign to each of five sales districts in a particular region. Each person is
likely to achieve a different sales volume in each district. Estimates of
these sales are shown in the table below. The company wishes to
maximize total sales volume. However, due to personnel rotation policies
which cannot be violated, it is not possible to assign salesperson B to
district 1 nor salesperson A to district 2. Solve the problem using the
Hungarian method and write a short report to the company indicating
which assignments should be made and the volume of sales likely to
result.
Salesperson
A
B
C
D
E
1
65
90
106
84
86
District
2
3
73 55
67 87
86 96
69 79
74 79
4
58
75
89
77
75
5
63
80
70
77
73
Determine how and to what extent, if at all, the personnel restrictions
affect the total estimated sales value.
10
11
12