University of Southern California
Daniel J. Epstein Department of Industrial and Systems Engineering
ISE 330: Introduction to Operations Research
Lecture 19 Outline: November 18, 2003
Instructor: Elaine Chew
Read:
H & L chapter 8.3
The Assignment Problem Definition
1.
2.
3.
4.
5.
The number of assignees and the number of tasks are the same, n.
Each assignee is to be assigned to exactly one task.
Each task is to be performed by exactly one assignee.
There is a cost cij associated with assignee i (i = 1,2,…,n) performing task j
(j = 1,2,…,n).
The objective is to determine how all n assignments should be made to
minimize the total cost.
Example: The quidditch assignment problem
Harry
Ron
Hermione
Chaser
13
15
5
Quidditch Position
Keeper
Beater
16
12
13
7
10
Seeker
11
20
6
The Assignment Problem Model:
E.C.
1
ISE 330: Lecture 19 Notes
http://www-classes.usc.edu/engr/ise/330
Does this look like a transportation problem? If so, what are the sources,
destinations, demands and supplies?
Do we need to solve the problem using integer programming?
Network Representation of the Assignment Problem:
E.C.
2
ISE 330: Lecture 19 Notes
http://www-classes.usc.edu/engr/ise/330
The cost table for the quidditch assignment problem:
Harry
Ron
Hermione
Chaser
13
15
5
Quidditch Position
Keeper
Beater
16
12
13
7
10
Seeker
11
20
6
Solution Procedure 1: Simplex Method
Dummy not necessary. Changes in the model…
Solution Procedure 2: Transportation Simplex Method
Optimal solution is x13=0, x14=1, x23=1, x31=1, x41=0, x42=1, x43=0.
Number of basic variables in a TSM solution is _________________
Number of non-degenerate basic variables is _________________
Number of degenerate basic variables is _________________
Drawbacks of using the transportation simplex method:
[1] Number of degenerate basic variables and wasted iterations.
[2] Does not exploit the additional structure of the assignment problem.
E.C.
3
ISE 330: Lecture 19 Notes
http://www-classes.usc.edu/engr/ise/330
Example: BETTER BROOMS COMPANY
Quidditch Team (unit cost)
Heidelberg
Quiberon
Woolongong Tchamba
Harriers
Quafflepunchers
Warriors
Charmers Supply
1
41
27
28
24
75
Broomery 2
40
29
23
75
3
37
30
27
21
45
Requirement
20
30
30
40
Option 1: Allow each team to be supplied by more than one broomery.
Option 2: Prohibit each team from being supplied by more than one broomery.
Solution Method I:
1
Broomery 2
3
Requirement
1
41
40
37
20
Quidditch Team (unit cost)
2
3
4
27
28
24
29
23
30
27
21
30
30
40
Supply
75
75
45
Optimal Solution: x12 = 30, x13 = 30, x15 = 15, x24 = 15, x25 = 60, x31 = 20, x34 = 25.
Total Cost, Z = $3260.
Solution Method II:
1
Quidditch Team Assignment
2
3
4
1a
1b
Broomery 2a
2b
3
Optimal Solution: Broomery 1 serves Teams 2 and 3, Broomery 2 serves Team 1,
Broomery 3 serves Team 4.
E.C.
4