Assignment problems

1. Assignment problems
Rahma TLILI ,Mohamed Amine ZAAG
15 November 2016

2. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 2/37

3. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 3/37

4. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
In the world of trade, Business organizations are confronting
the conflicting need for optimal utilization of their limited
resources.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 4/37

5. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
In the world of trade, Business organizations are confronting
the conflicting need for optimal utilization of their limited
resources.
The optimal deployment can offer for the company
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 4/37

6. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
In the world of trade, Business organizations are confronting
the conflicting need for optimal utilization of their limited
resources.
The optimal deployment can offer for the company
Minimal Cost
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 4/37

7. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
In the world of trade, Business organizations are confronting
the conflicting need for optimal utilization of their limited
resources.
The optimal deployment can offer for the company
Minimal Cost
Minimal time for accomplishing jobs
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 4/37

8. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
In the world of trade, Business organizations are confronting
the conflicting need for optimal utilization of their limited
resources.
The optimal deployment can offer for the company
Minimal Cost
Minimal time for accomplishing jobs
Maximum Benefit. . . .
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 4/37

9. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
In the world of trade, Business organizations are confronting
the conflicting need for optimal utilization of their limited
resources.
The optimal deployment can offer for the company
Minimal Cost
Minimal time for accomplishing jobs
Maximum Benefit. . . .
When the number of workers staff is limited we can easily find
an optimal assignment of jobs for each individual.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 4/37

10. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
But what about the case we have a huge number of tasks to
be done and respectively big number of employees ?
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 5/37

11. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
But what about the case we have a huge number of tasks to
be done and respectively big number of employees ?
When the number increases the resolution of our problem will
be more difficult and complicate to resolve.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 5/37

12. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Introduction
But what about the case we have a huge number of tasks to
be done and respectively big number of employees ?
When the number increases the resolution of our problem will
be more difficult and complicate to resolve.
We have to find such an assignment by which we can get
maximum profit on minimum investment. Such problems are
known as ”assignment problems” .
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 5/37

13. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 6/37

14. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Example
A company has three men available for work on Three
separate jobs. Only one man can work on any one job. The
objective is to assign men to jobs such that the total cost of
assignment is minimum.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 7/37

15. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Example
A company has three men available for work on Three
separate jobs. Only one man can work on any one job. The
objective is to assign men to jobs such that the total cost of
assignment is minimum.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 7/37

16. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 8/37

17. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Mathematically, we can express the problem as follows:
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 8/37

18. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Mathematically, we can express the problem as follows:
MinZ(cost) =
n
i=1
n
j=1
cij xij
cij Cost of assigning jth work to ith person
Where
xij = 1 if i person is assigned j work
xij = 0 if i person is not assigned j work
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 8/37

19. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 9/37

20. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
We can represent the table above as a cost matrix.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 9/37

21. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
We can represent the table above as a cost matrix.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 9/37

22. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
We can represent the table above as a cost matrix.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 9/37

23. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 10/37

24. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
First one possible assignment.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 10/37

25. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
First one possible assignment.
The total cost of this assignment is : 250 + 600 + 250 = 1100
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 10/37

26. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 11/37

27. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Second possible assignment.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 11/37

28. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Second possible assignment.
The total cost of this assignment is :250 + 350 + 400 = 1000
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 11/37

29. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
After checking all six possible assignments we can determine that
the optimal one is the following.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 12/37

30. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
After checking all six possible assignments we can determine that
the optimal one is the following.
The total cost of this assignment is :400 + 350 + 200 = 950
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 12/37

31. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Trial and error works well enough for this problem, but
suppose you had ten jobs assigned to ten persons? How many
trials would this take?
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 13/37

32. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Trial and error works well enough for this problem, but
suppose you had ten jobs assigned to ten persons? How many
trials would this take?
There are n! ways of assigning n resources to n tasks.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 13/37

33. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Trial and error works well enough for this problem, but
suppose you had ten jobs assigned to ten persons? How many
trials would this take?
There are n! ways of assigning n resources to n tasks.
That means that as n gets large, we have too many trials to
consider
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 13/37

34. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 14/37

35. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Example
Mathematic model
Complexity
Complexity
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 14/37

36. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 15/37

37. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Exact solution methods
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 16/37

38. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Exact solution methods
Maximum matching algorithm for
weighted bipartite graphs
(MWM)
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 16/37

39. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Exact solution methods
Maximum matching algorithm for
weighted bipartite graphs
(MWM)
Successive shortest path method
(SSP)
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 16/37

40. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Exact solution methods
Maximum matching algorithm for
weighted bipartite graphs
(MWM)
Successive shortest path method
(SSP)
Hungarian method
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 16/37

41. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Comparison between methods
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 17/37

42. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Exact solution methos
Comparison
Comparison between methods
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 17/37

43. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 18/37

44. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Presentation
This method was developed by american mathematician
Harold William Kuhn (1925–2014)
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 19/37

45. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Presentation
This method was developed by american mathematician
Harold William Kuhn (1925–2014)
He uses to refer to his method as ”Hungarian method” (This
method was started with two mathematicians: D´enes K˝onig
and Jen˝o Egerv´ary from Hungary)
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 19/37

46. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Presentation
This method was developed by american mathematician
Harold William Kuhn (1925–2014)
He uses to refer to his method as ”Hungarian method” (This
method was started with two mathematicians: D´enes K˝onig
and Jen˝o Egerv´ary from Hungary)
An algorithm which finds an optimal assignment for a given
cost matrix.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 19/37

47. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 20/37

48. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
Theorem
If a constant is added (or subtracted) to every element of any row
(or column) of the cost matrix [cij ] in an assignment problem then
an assignment which minimizes the total cost for the new matrix
will also minimize the total cost matrix.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 20/37

49. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
1 Subtract the smallest entry in each row from all the entries of
its row
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 21/37

50. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
1 Subtract the smallest entry in each row from all the entries of
its row
2 Subtract the smallest entry in each column from all the
entries of its column
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 21/37

51. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
1 Subtract the smallest entry in each row from all the entries of
its row
2 Subtract the smallest entry in each column from all the
entries of its column
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 21/37

52. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
3 Examine the rows successively until a row with a single zero is
found. Make this zero in a square row and cross off (X) all
other zeros in its column.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 22/37

53. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
3 Examine the rows successively until a row with a single zero is
found. Make this zero in a square row and cross off (X) all
other zeros in its column.
4 Repeat the procedure respectively for each column of the
reduced matrix.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 22/37

54. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
5 If the number of assignment (squares) is equal to n (the order
of the cost matrix),an optimum solution is reached, else go
to the next step
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 23/37

55. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
5 If the number of assignment (squares) is equal to n (the order
of the cost matrix),an optimum solution is reached, else go
to the next step
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 23/37

56. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
6 Draw the minimum number of vertical and horizontal lines
necessary to cover all the zeros in the reduced matrix obtained
from last step.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 24/37

57. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
6 Draw the minimum number of vertical and horizontal lines
necessary to cover all the zeros in the reduced matrix obtained
from last step.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 24/37

58. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
7 Select the smallest element from all the uncovered elements.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 25/37

59. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
7 Select the smallest element from all the uncovered elements.
8 Subtract this smallest element from all the uncovered
elements and add it to the elements which lie at the
intersection of two lines.We obtain another reduced matrix
for fresh assignment.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 25/37

60. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
7 Select the smallest element from all the uncovered elements.
8 Subtract this smallest element from all the uncovered
elements and add it to the elements which lie at the
intersection of two lines.We obtain another reduced matrix
for fresh assignment.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 25/37

61. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
7 Select the smallest element from all the uncovered elements.
8 Subtract this smallest element from all the uncovered
elements and add it to the elements which lie at the
intersection of two lines.We obtain another reduced matrix
for fresh assignment.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 26/37

62. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
We repeat previous steps and we obtain a new number of
assignments which is equal to the number of rows and
columns, this is the optimal solution.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 27/37

63. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Steps
We repeat previous steps and we obtain a new number of
assignments which is equal to the number of rows and
columns, this is the optimal solution.
The optimal assignment = A1 + B4 + C2 + D3
The total cost of assignment: 20 + 17 + 24 + 17 = 78.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 27/37

64. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Unbalanced assignment problem
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 28/37

65. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Unbalanced assignment problem
It is an assignment problem where the number of persons is
not equal to the number of jobs.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 28/37

66. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Unbalanced assignment problem
It is an assignment problem where the number of persons is
not equal to the number of jobs.
If the number of persons is less than the number of jobs then
we introduce one or more dummy persons (rows) with zero
values to make the assignment problem balanced.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 28/37

67. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Presentation
Algorithm steps
Unbalanced assignment problem
Unbalanced assignment problem
It is an assignment problem where the number of persons is
not equal to the number of jobs.
If the number of persons is less than the number of jobs then
we introduce one or more dummy persons (rows) with zero
values to make the assignment problem balanced.
Likewise, if the number of jobs is less than the number of
persons then we introduce one or more dummy jobs
(columns) with zero values to make the assignment problem
balanced.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 28/37

68. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 29/37

69. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Application areas
Match jobs to machines
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 30/37

70. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Application areas
Match jobs to machines
Match personnel to tasks
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 30/37

71. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Application areas
Match jobs to machines
Match personnel to tasks
Determining positions on a team
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 30/37

72. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Application areas
Match jobs to machines
Match personnel to tasks
Determining positions on a team
...
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 30/37

73. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Non-obvious applications
Vehicle routing
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 31/37

74. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Non-obvious applications
Vehicle routing
Multiple object tracking
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 31/37

75. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Non-obvious applications
Vehicle routing
Multiple object tracking
Signal processing
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 31/37

76. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Non-obvious applications
Vehicle routing
Multiple object tracking
Signal processing
Assigning cells to switches in cellular mobile networks.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 31/37

77. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Natural application
Non-obvious application
Virtual output queueing ”VOQ”
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 32/37

78. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Outline
1 Introduction
2 Formulation of the problem
3 Solving the problem
4 Hungarian method
5 Application areas
6 Conclusion & perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 33/37

79. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Conclusion
Assignment problem is a special type of linear programming
problem which deals with the allocation of the various
resources to the various activities on one to one basis.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 34/37

80. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Conclusion
Assignment problem is a special type of linear programming
problem which deals with the allocation of the various
resources to the various activities on one to one basis.
It does it in such a way that the cost or time involved in the
process is minimum and profit or sale is maximum.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 34/37

81. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Conclusion
Assignment problem is a special type of linear programming
problem which deals with the allocation of the various
resources to the various activities on one to one basis.
It does it in such a way that the cost or time involved in the
process is minimum and profit or sale is maximum.
Such problems can be solved by simplex method or by
transportation method but assignment model gives a simpler
approach for these problems.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 34/37

82. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Conclusion
Assignment problem is a special type of linear programming
problem which deals with the allocation of the various
resources to the various activities on one to one basis.
It does it in such a way that the cost or time involved in the
process is minimum and profit or sale is maximum.
Such problems can be solved by simplex method or by
transportation method but assignment model gives a simpler
approach for these problems.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 34/37

83. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Perspectives
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 35/37

84. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Perspectives
Future work
Advanced Heuristics and Meta-heuristics
More exact solution methods
Expand algorithms to solve variations of the assignment
problems.
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 35/37

85. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
References
http://s3.amazonaws.com/ppt-download/
assignmentproblems-140831020535-phpapp02.pdf
http://www.csir.co.za/dpss/ledger/docs/Grant_
Dieman.ppt
http://s3.amazonaws.com/ppt-download/
lesson33-assignment-problem-1197479706102836-5.
ppt
http://www.math.harvard.edu/archive/20_spring_05/
handouts/assignment_overheads.pdf
http://s3.amazonaws.com/ppt-download/
assignmentproblem-130402010013-phpapp02.pptx
Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 36/37

86. Introduction
Formulation of the problem
Solving the problem
Hungarian method
Application areas
Conclusion & perspectives
Thank you for your attention
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Rahma TLILI ,Mohamed Amine ZAAG Assignment problems 37/37