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 sixhundred
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
sixhundred
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...

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