Operations Research
The assignment problem – exercise.
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
The assignment problem – exercise
The assignment problems of minimization type – examples
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 1:
Solve the minimization assignment problem
0
5 6
9
B 5 5
7
B
@ 6 7 10
9 11 13
Michal Šmerek
with a matrix rates
1
10
8 C
C.
10 A
14
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 1:
0
0
B 0
B
Xopt = @
0
1
1
0
0
0
0
1
0
0
1
0
0 C
C,
1 A
0
zmin = 32.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 2:
Solve the minimization assignment
0
9
B 7
B
@ 7
9
problem with a matrix rates
1
7 6 3
9 4 3 C
C.
8 5 7 A
6 1 2
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 2:
0
0
B 0
B
Xopt = @
1
0
1
0
0
0
0
0
0
1
1
0
1 C
C,
0 A
0
zmin = 18.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 3:
Solve the minimization assignment
0
9
B 7
B
@ 7
6
problem with a matrix rates
1
2 7 7
5 7 8 C
C.
5 3 7 A
2 9 9
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 3:
0
Xopt,1
0
B 0
B
=@
0
1
1
0
0
0
0
0
1
0
1
0
0
0
B
1 C
C , Xopt,2 = B 1
@ 0
0 A
0
0
Michal Šmerek
0
0
0
1
0
0
1
0
1
1
0 C
C,
0 A
0
Linear programming
zmin = 19.
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 4:
Solve the minimization assignment
0
3
B 3
B
@ 6
5
problem with a matrix rates
1
10 5 5
5
3 2 C
C.
8 10 6 A
9
2 2
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 4:
0
1
B 0
Xopt = B
@ 0
0
0
0
1
0
0
0
0
1
1
0
1 C
C,
0 A
0
zmin = 15.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
The assignment problems of maximization type – examples
Example 1:
Solve the maximization assignment problem
0
5 6
9
B 5 5
7
B
@ 6 7 10
9 11 13
Michal Šmerek
with a matrix rates
1
10
8 C
C.
10 A
14
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 1:
0
0
B 1
B
Xopt = @
0
0
0
0
0
1
0
0
1
0
1
1
0 C
C,
0 A
0
zmax = 36.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 2:
Solve the maximization assignment
0
9
B 7
B
@ 7
9
problem with a matrix rates
1
7 6 3
9 4 3 C
C.
8 5 7 A
6 1 2
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 2:
0
0
B 0
B
Xopt = @
0
1
0
1
0
0
1
0
0
0
1
0
0 C
C,
1 A
0
zmax = 31.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 3:
Solve the maximization assignment
0
9
B 7
B
@ 7
6
problem with a matrix rates
1
2 7 7
5 7 8 C
C.
5 3 7 A
2 9 9
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 3:
0
1
B 0
B
Xopt = @
0
0
0
0
1
0
0
0
0
1
1
0
1 C
C,
0 A
0
zmax = 31.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 4:
Solve the maximization assignment problem
0
3 10 5
B 3 5
3
B
@ 6 8 10
5 9
2
Michal Šmerek
with a matrix rates
1
5
2 C
C.
6 A
2
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 4:
0
Xopt,1
0
B 0
B
=@
0
1
1
0
0
0
0
0
1
0
1
0
0
0
B
1 C
C , Xopt,2 = B 1
@ 0
0 A
0
0
Michal Šmerek
0
0
0
1
0
0
1
0
1
1
0 C
C,
0 A
0
Linear programming
zmax = 27.
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
The assignment problems of both types – examples
Example 1:
Solve the assignment problem with
0
2
B 2
B
@ 1
4
a matrix rates
1
3 11 4
5 12 6 C
C.
5 9 6 A
6 9 7
Determine all optimal solution of the problem if the problem is:
a) minimization,
b) maximization.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 1:
0
B 0
=B
@ 1
0
1
0
0
0
0
0
0
1
1
0
1 C
C,
0 A
0
0
1
B 0
=B
@ 0
0
0
0
1
0
0
1
0
0
1
0
0 C
C,
0 A
1
0
0
1
0
0
1
0
0
0
1
0
0 C
C,
1 A
0
0
a) Xopt,1
b) Xopt,1
Xopt,3
0
B 0
=B
@ 0
1
0
B 0
=B
@ 1
0
0
1
0
0
0
0
0
1
1
1
0 C
C,
0 A
0
0
1
B 0
=B
@ 0
0
0
0
0
1
0
1
0
0
1
0
0 C
C,
1 A
0
0
0
0
1
0
1
0
0
0
1
0
1 C
C,
0 A
0
0
Xopt,2
Xopt,2
Xopt,4
Michal Šmerek
0
B 0
=B
@ 0
1
Linear programming
zmin = 19,
zmax = 26.
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 2:
Solve the assignment problem with
0
6
B 5
B
@ 7
5
a matrix rates
1
1 2 11
2 3 12 C
C.
2 3 12 A
2 2 8
Determine all optimal solution of the problem if the problem is:
a) minimization,
b) maximization.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 2:
0
B 1
=B
@ 0
0
1
0
0
0
0
0
1
0
1
0
0 C
C,
0 A
1
0
1
B 0
=B
@ 0
0
0
0
0
1
0
1
0
0
1
0
0 C
C,
1 A
0
0
0
0
0
1
1
0
0
0
1
0
1 C
C,
0 A
0
0
a) Xopt,1
b) Xopt,1
Xopt,3
0
B 0
=B
@ 1
0
0
B 1
=B
@ 0
0
0
0
1
0
1
0
0
0
1
0
0 C
C,
0 A
1
0
1
B 0
=B
@ 0
0
0
0
0
1
0
0
1
0
1
0
1 C
C,
0 A
0
0
0
0
0
1
0
1
0
0
1
1
0 C
C,
0 A
0
0
Xopt,2
Xopt,2
Xopt,4
Michal Šmerek
0
B 0
=B
@ 1
0
Linear programming
zmin = 17,
zmax = 23.
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Example 3:
Solve the assignment problem with a
0
9 5
B 8 8
B
B 7 7
B
@ 9 8
6 5
matrix rates
1
7 1 3
3 6 6 C
C
3 3 5 C
C.
4 6 7 A
8 7 3
Determine all optimal solution of the problem if the problem is:
a) minimization,
b) maximization.
Michal Šmerek
Linear programming
The assignment problems of minimization type – examples
The assignment problems of maximization type – examples
The assignment problems of both types – examples
Solution 3:
0
B
B
a) Xopt = B
B
@
0
B
B
b) Xopt = B
B
@
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
1
C
C
C,
C
A
C
C
C,
C
A
Michal Šmerek
zmin = 22,
zmax = 37.
Linear programming
Operations Research
The assignment problem – exercise.