This preview shows pages 1–5. Sign up to view the full content.
2/22
Assignment Problems
Example
Machineco has four jobs to be completed. Each machine must be
assigned to complete one job. The time required to setup each
machine for completing each job is shown in the following table
(also called the
cost matrix
):
Time (Hours)
Job 1
Job 2
Job 3
Job 4
Machine 1
14
5
8
7
Machine 2
2
12
6
5
Machine 3
7
8
3
9
Machine 4
2
4
6
10
Machinco wants to minimize the total setup time needed to
complete the four jobs.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 3/22
Assignment Problems
±
In general, an assignment problem is a balanced transportation
problem, in which all supplies and demands are equal to 1.
±
This is a special subclass of transportation problems that can
be solved more eﬃciently than the general transportation
problems.
Next, we will develop a linear programming model for our example.
±
The decision variables:
x
ij
=
±
1
,
if machine
i
is assigned to meet the demands of job
j
0
,
otherwise
.
4/22
Assignment Problems
Based on the cost matrix, Machinco’s problem can be formulated
as follows (for
i
,
j
= 1
,
2
,
3
,
4):
min
z
= 14
x
11
+ 5
x
12
+ 8
x
13
+ 7
x
14
+2
x
21
+ 12
x
22
+ 6
x
23
+ 5
x
24
+7
x
31
+ 8
x
32
+ 3
x
33
+ 9
x
34
+2
x
41
+
x
42
+ 6
x
43
+ 10
x
44
s
.
t
.
x
11
+
x
12
+
x
13
+
x
14
= 1
x
21
+
x
22
+
x
23
+
x
24
= 1
x
31
+
x
32
+
x
33
+
x
34
= 1
x
41
+
x
42
+
x
43
+
x
44
= 1
x
11
+
x
21
+
x
31
+
x
41
= 1
x
12
+
x
22
+
x
32
+
x
42
= 1
x
13
+
x
23
+
x
33
+
x
43
= 1
x
14
+
x
24
+
x
34
+
x
44
= 1
x
ij
= 0
or
x
ij
= 1
,
i
,
j
= 1
, . . . ,
4
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 5/22
Assignment Problems
±
Note that we have the integrality constraints:
x
ij
= 0
or
x
ij
= 1
,
i
,
j
= 1
, . . . ,
4
±
However, if we remove the integrality constraints and solve
the resulting transportation problem with the transportation
simplex, we will obtain an integer solution.
±
Indeed, all the input data are integer, and we use only
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/14/2010 for the course ESI 6314 taught by Professor Vladimirlboginski during the Fall '09 term at University of Florida.
 Fall '09
 VLADIMIRLBOGINSKI

Click to edit the document details