Student
The Assignment Problem and the Hungarian Method
1
Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are in Austin, TX;
Boston, MA; and Chicago, IL. You want them to fly to three other cities: Denver, CO; Edmonton, Alberta; and Fargo,
ND. The table below shows the cost of airplane tickets in dollars between these cities.
From \ To
Denver
Edmonton
Fargo
Austin
250
400
350
Boston
400
600
350
Chicago
200
400
250
Where should you send each of your salespeople in order to minimize airfare?
2
We can represent the table above
250 400
400 600
200 400
3
as a cost matrix.
350
350
250
Let’s look at one possible assignment.
250
400
350
400
600
350
250
200
400
The total cost of this assignment is
$250 + $600 + $250 = $1100.
4
Here’s another possible assignment.
250
400
350
400
600
350
200
400
250
The total cost of this assignment is
$250 + $350 + $400 = $1000.
5
After checking all six possible assignments we can determine that the optimal one is the following.
250
400
350
400
350
600
200
400
250
The total cost of this assignment is
$400 + $350 + $200 = $950.
Thus your salespeople should travel from Austin to
Edmonton, Boston to Fargo, and Chicago to Denver.
6
Trial and error works well enough for this problem, but suppose you had ten salespeople flying to ten cities? 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.
7
7
6
5
4
3
2
1
n
2
3
4
8
5
6
7
40
30
20
n2
10
n
1
2
3
4
9
5
6
7
1000
800
600
en
400
n2
200
n
1
2
3
1
Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are in Austin, TX;
Boston, MA; and Chicago, IL. You want them to fly to three other cities: Denver, CO; Edmonton, Alberta; and Fargo,
ND. The table below shows the cost of airplane tickets in dollars between these cities.
From \ To
Denver
Edmonton
Fargo
Austin
250
400
350
Boston
400
600
350
Chicago
200
400
250
Where should you send each of your salespeople in order to minimize airfare?
2
We can represent the table above
250 400
400 600
200 400
3
as a cost matrix.
350
350
250
Let’s look at one possible assignment.
250
400
350
400
600
350
250
200
400
The total cost of this assignment is
$250 + $600 + $250 = $1100.
4
Here’s another possible assignment.
250
400
350
400
600
350
200
400
250
The total cost of this assignment is
$250 + $350 + $400 = $1000.
5
After checking all six possible assignments we can determine that the optimal one is the following.
250
400
350
400
350
600
200
400
250
The total cost of this assignment is
$400 + $350 + $200 = $950.
Thus your salespeople should travel from Austin to
Edmonton, Boston to Fargo, and Chicago to Denver.
6
Trial and error works well enough for this problem, but suppose you had ten salespeople flying to ten cities? 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.
7
7
6
5
4
3
2
1
n
2
3
4
8
5
6
7
40
30
20
n2
10
n
1
2
3
4
9
5
6
7
1000
800
600
en
400
n2
200
n
1
2
3