An-Najah N. University
Faculty of Engineering and Information Technology
Department of Management Information systems
Operations Research and Applications MIS:10676213
Prepared by Mr.Maher Abubaker
Fall 2015/2016
Resources
Daniel J. Epstein Department of Industrial and Systems Engineering University of Southern California
Andrew and Erna Viterbi School of Engineering
http://www.eecs.qmul.ac.uk/~eniale/teaching/ise330/index.html
Operations Research: An Introduction, 9/EHamdy A. Taha, University of ArkansasISBN-10:
013255593X ISBN-13: 9780132555937©2011 • Prentice Hall • Cloth, 832 ppPublished 08/29/2010
http://www.pearsonhighered.com/educator/product/Operations-Research-AnIntroduction/9780132555937.page#downlaoddiv
™
INFORMS – www.informs.org ™
ORMS - www.lionhrtpub.com/ORMS.shtml ™
Science of Better - www.scienceofbetter.org
The Assignment Problem
A special case of the transportation
problem which is a special case of a
linear program
The Problem
There are n workers and n tasks to be performed.
The time it takes worker i to perform task j is cij.
Which task should be assigned to which workers?
I want the easy task.
The Model
n
n
Min z    cij xij
i 1 j 1
s. t .
n
x
1
j  1,2,..., n
1
i  1,2,..., n
xij  0,1
for all i , j
ij
i 1
n
x
ij
j 1
animated
This is just
the transportation
problem with the
right hand side
values equal to one!
Some Applications
These are terrific
applications.
•
•
•
•
•
•
•
workers to tasks
jobs to machines
facilities to locations
Truck drivers to customer pick-up points
Umpire crews to baseball games
Judges to court dockets
State inspectors to construction sites
A combinatorial problem
If there are n workers and n tasks there are
n! (factorial) possible assignments.
Example: Workers are Al, Art, Alice, and Ann. There
are four tasks: 1,2,3, & 4.
4
Al
3
Art
2
1
Alice Ann
If n = 10 then 10! = 3,628,800
=
4!
An Example - assign a construction project
(building) to a contractor
Building
Contractor
1
2
3
4
A
48
48
50
44
B
56
60
60
68
C
96
94
90
85
D
42
44
54
46
Bids (in $10,000)
The Algorithm (Flood’s or the Hungarian Method)
• 1. Subtract the smallest cost element in each row from
every element in that row.
• 2. Subtract the smallest cost element in each column from
every element in that column.
• 3. Test for optimality by drawing the minimum number of
lines that will cover every zero cell (no diagonal lines). If
the minimum = n, a feasible assignment involving only
zero cells is possible. Go to step 5.
• 4. Select the smallest element not having a line through it.
Subtract this amount from all elements not covered by a
line;and add this amount to all elements at the intersection
of lines.Go to step 3.
• 5. Solution is optimum; make assignments using zero cells
so that all constraints are satisfied.
n
n
z    cij xij
I bet it works
by magic.
Some Magic
i 1 j 1
Subtract a constant ‘a’ from row k:
n
n
n
n
n
n
z '    cij xij   (ckj  a ) xkj    cij xij  a  xkj
i 1 j 1
ik
j 1
n
i 1 j 1
j 1
n
   cij xij  a  z  a
i 1 j 1
n
since
x
j 1
kj
1
animated
Let’s solve the problem-1!
1
2
3
4
A
48
48
50
44
B
56
60
60
68
C
96
94
90
85
D
42
44
54
46
Let’s solve the problem-2!
Subtract 44 from row 1
1
2
3
4
A
4
4
6
0
B
56
60
60
68
C
96
94
90
85
D
42
44
54
46
Let’s solve the problem-3!
Subtract 56 from row 2
1
2
3
4
A
4
4
6
0
B
0
4
4
12
C
96
94
90
85
D
42
44
54
46
Let’s solve the problem-4!
Subtract 85 from row 3
and 42 from row 4
1
2
3
4
A
4
4
6
0
B
0
4
4
12
C
11
9
5
0
D
0
2
12
4
Let’s solve the problem-5!
Is there a feasible solution
using the zero cells?
Subtract 2 from column 2
and 4 from column 3
animated
1
2
3
4
A
4
2
2
0
B
0
2
0
12
C
11
7
1
0
D
0
0
8
4
3=4
Let’s solve the problem-6!
Minimum uncovered element
animated
1
2
3
4
A
4
2
2
0
B
0
2
0
12
C
11
7
1
0
D
0
0
8
4
Let’s solve the problem-7!
1
2
3
4
A
4-1=3
2-1=1
2-1=1
0
B
0
2
0
12+1=13
C
11-1=10
7-1=6
1-1=0
0
D
0
0
8
4+1=5
animated
Let’s solve the problem-8!
1
2
3
4
A
3
1
1
0
B
0
2
0
13
C
10
6
0
0
D
0
0
8
5
Optimal assignment is now possible
using the zero cells!
animated
Let’s solve the problem-9!
1
2
3
4
A
3
1
1
X 0
B
0
2
0
13
C
10
6
0
0
D
0
0
8
animated
X
X
X
5
Another Example
The Match Maker, a computerized dating service that attempts
to bring two compatible people together, has to match the
following individuals: Mandy, Mollie, and Martha with
Bill, Bob, Bruno, and Bruce. The ladies have ranked each man
on a scale of 1 to 10 with the higher number being the more preferred.
The Rankings-1
Bill
Bob
Bruno
Bruce
Mandy
7
5
10
6
Mollie
8
6
9
5
Martha
9
7
8
4
The Rankings-2
Bill
Bob
Bruno
Bruce
Mandy
7
5
10
6
Mollie
8
6
9
5
Martha
9
7
8
4
Dummy
0
0
0
0
Convert to a minimization problem
Bill
Bob
Bruno
Bruce
Mandy
10-7=3
10-5=5
10-10=0
10-6=4
Mollie
9-8=1
9-6=3
9-9=0
9-5=4
Martha
9-9=0
9-7=2
9-8=1
9-4=5
Dummy
0
0
0
0
Test for optimality-1
Bill
Bob
Bruno
Bruce
Mandy
3
5
0
4
Mollie
1
3
0
4
Martha
0
2
1
5
Dummy
0
0
0
0
animated
Test for optimality-2
Bill
Bob
Bruno
Bruce
Mandy
3
5
0
4
Mollie
1
3
0
4
Martha
0
2
1
5
Dummy
0
0
0
0
Minimum uncovered cell
animated
Test for optimality-3
Bill
Bob
Bruno
Bruce
Mandy
3-1=2
5-1=4
0
4-1=3
Mollie
1-1=0
3-1=2
0
4-1=3
Martha
0
2
1+1=2
5
Dummy
0
0
0+1=1
0
animated
Test for optimality-4
Bill
Bob
Bruno
Bruce
Mandy
2
4
0
3
Mollie
0
2
0
3
Martha
0
2
2
5
Dummy
0
0
1
0
animated
Minimum uncovered cell
Test for optimality-5
Bill
Bob
Bruno
Bruce
Mandy
2
4-2=2
0
3-2=1
Mollie
0
2-2=0
0
3-2=1
Martha
0
2-2=0
2
5-2=3
Dummy
0+2=2
0
1+2=3
0
animated
Test for optimality-6
Bill
Bob
Bruno
Bruce
Mandy
2
2
0
1
Mollie
0
0
0
1
Martha
0
0
2
3
Dummy
2
0
3
0
animated
Test for optimality-7
Bill
Bob
Bruno
Bruce
Mandy
2
2
0
1
Mollie
0
0
0
1
Martha
0
0
2
3
0
3
0
X
X
X
Dummy
animated
2
X
7! = 5040
Scheduling Umpires
Night game to
a day game
Flight scheduling
problems
Need a travel day off
KC
MIN
CHI
MKE
DET
TOT
NY
SEA(7)
M
1399
M
1694
1939
2124
2421
OAK(3)
1498
1589
M
1845
2079
2286
2586
CAL(2)
1363
1536
M
1756
1979
2175
2475
TEX(6)
M
853
798
843
982
1186
1383
MIN(5)
394
0
334
297
528
780
1028
CHI(4)
403
334
0
74
235
430
740
TOR(1)
M
M
M
M
206
M
366
animated
No crew assigned to same team more than 2 series
The Assignment Problem
This has been another delightful OR
experience that both invigorates and
excites the mind. Don’t you agree?
return