The Project Problem

and

the Hungarian Method

1

Example 1: You act as a sales manager for a toy

maker, and you currently have three salesmen on

the street meeting customers. Your salesmen are in Austin, TX; Boston, MA; and Chicago, ELLE. You desire them to п¬‚y to three

additional cities: Hawaii, CO; Edmonton, Alberta; and Fargo,

ND. The desk below reveals the cost of airplane tickets in

dollars among these towns.

From \ To

Denver colorado

Edmonton

Fargo

Austin

two hundred fifity

400

350

Boston

four hundred

600

three hundred and fifty

Chicago

2 hundred

400

250

Where in the event you send every of your salespeople in order to

decrease airfare?

a couple of

We can represent the desk above

пЈ®

250 four hundred

пЈЇ

пЈЇ

пЈЇ400 six-hundred

пЈ°

200 400

3

as a cost matrix.

пЈ№

350

пЈє

пЈє

350пЈє

пЈ»

two hundred and fifty

Let's take a look at one conceivable assignment.

пЈ®

250

400

350

пЈЇ

пЈЇ

пЈЇ 400

600

350

пЈ°

250

200

400

The entire cost of this kind of assignment is definitely

$250 & $600 + $250 = $1100.

5

пЈ№

пЈє

пЈє

пЈє

пЈ»

This another possible assignment.

пЈ®

250

500

350

пЈЇ

пЈЇ

пЈЇ 400

six-hundred

350

пЈ°

200

4 hundred

250

The entire cost of this kind of assignment can be

$250 + $350 + $400 = $1000.

a few

пЈ№

пЈє

пЈє

пЈє

пЈ»

After checking every six possible assignments we can determine the fact that optimal is the following.

пЈ®

пЈ№

two hundred fifty

400

three hundred and fifty

пЈЇ

пЈє

пЈЇ

пЈє

пЈЇ 500

350 пЈє

600

пЈ°

пЈ»

200

400

250

The total cost of this job is

$400 + $350 + $200 = $950.

Thus your salespeople ought to travel coming from Austin to

Edmonton, Boston to Fargo, and Chicago to Denver.

6

Learning from your errors works well enough for this problem, but

presume you had 10 salespeople п¬‚ying to 10 cities? How

many trial offers would this kind of take?

You will find n! methods of assigning and resources to n tasks.

That means that as and gets large, we have too many trials

to consider.

7

7

6

5

some

3

a couple of

1

n

2

several

4

almost eight

5

six

7

45

30

twenty

n2

10

n

you

2

3

4

9

5

6

7

1000

800

six hundred

en

500

n2

two hundred

n

1

2

three or more

10

four

5

six

7

5000

4000

n

3000

en

2000

n2

1000

n

1

a couple of

3

14

4

five

6

several

Theorem: If a number is usually added to or perhaps subtracted coming from all

in the entries of any one row or line of a expense matrix,

in that case on optimal assignment for the causing cost matrix

is also a great optimal task for the first cost matrix.

12

The Hungarian Technique: The following algorithm applies these theorem to a given in Г— in cost matrix to п¬Ѓnd an maximum assignment. Step one. Subtract the smallest entry in each row from each of the entries of its line.

Step 2. Take away the smallest entrance in every single column by all the entries of it is column.

3. Draw lines through suitable rows and columns in order that all the absolutely no entries off the cost matrix happen to be covered as well as the minimum volume of such lines is used.

Step four. Test pertaining to Optimality: (i) If the bare minimum number of covering up lines is definitely n, a great optimal job of zeros is possible and are п¬Ѓnished. (ii) In the event the minimum number of covering lines is less than in, an optimum assignment of zeros is usually not yet feasible. In that case, check out Step 5. Step 5. Determine the smallest entry not covered by any line. Subtract this admittance from every uncovered row, and then put it with each covered steering column. Return to 3.

13

Case in point 1: You work as a sales manager for a plaything

manufacturer, and you simply currently have 3 salespeople upon

the road conference buyers. The salespeople happen to be in Austin, TX; Boston, MA; and Chicago, IL. You want these to п¬‚y to three

other towns: Denver, CO; Edmonton, Alberta; and Fargo,

ND. The table below shows the price tag on airplane entry pass in

dollars between these kinds of cities.

From \ To

Denver

Edmonton

Fargo

Austin texas

250

500

350

Boston

400

six hundred

350

Chicago

200

4 hundred

250

Exactly where should you send each of your salespeople to be able to

minimize airfare?

14

Step one. Subtract two hundred fifty

200 from Row three or more.

пЈ®

250 400

пЈЇ

пЈЇ

пЈЇ400 600

пЈ°

200 four hundred

from Line 1, 350 from...