Solution methodologies for the classical
assignment problem
Grant van Dieman
Friday 30th November 2007
Supervisor: Prof. JH van Vuuren
Co-supervisor: Mr JN Roux
Overview
 The classical assignment problem
 Exact Solution methods
 A maximum matching algorithm
 Successive shortest path method
 Hungarian method
 Greedy heuristics
 Comparison
 Future work
Slide 2
The classical assignment problem
 Votaw and Orden (1952)
 Assumptions
xij is 1 if assignee i is assigned to task j and 0 otherwise
The assignment problem is NP complete (Lloyd and
Witzenhausen (1986))
Slide 3
The Weapon Target Assignment Problem
 Flood (1957)
 Vj : priority of eliminating target j.
 qij : is the survival probability of target j if it is
engaged by weapon i.
 xij =1 if weapon i engage target j and 0 otherwise
Slide 4
Overview
 The classical assignment problem
 Exact Solution methods
 A maximum matching algorithm
 Successive shortest path method
 Hungarian method
 Greedy heuristics
 Comparison
 Future work
Slide 5
A maximum matching algorithm for weighted
bipartite graphs (MWM)
V1 = {assignees}
G:
qij
V2 = {tasks}
Slide 6
A maximum matching algorithm for weighted
bipartite graphs (MWM)
V1 = {assignees}
M:
qij
V2 = {tasks}
Slide 7
Overview
 The classical assignment problem
 Exact Solution methods
 A maximum matching algorithm
 Successive shortest path method
 Hungarian method
 Greedy heuristics
 Comparison
 Future work
Slide 8
Successive shortest path algorithm
(SSP)
 Minimum cost flow algorithm
Minimize  cij xij
i,j E
subject to
x

 
j: j ,i E
ji

x

 
j: i , j E
ij
 bi 
xij  uij
xij  lij
i V ,
i, j  E ,
i, j   E.
 Why this algorithm can be used to solve the assignment
problem
 The value of xij will be binary
Slide 9
Overview
 The classical assignment problem
 Exact Solution methods
 Successive shortest path method
 A maximum matching algorithm
 Hungarian method
 Greedy heuristics
 Comparison
 Future work
Slide 10
Hungarian Method
 Kuhn(1955)
 Special algorithm for the assignment problem
 Construct reduced cost matrix
Slide 11
Overview
 The classical assignment problem
 Exact Solution methods
 Successive shortest path method
 A maximum matching algorithm
 Hungarian method
 Greedy heuristics
 Comparison
 Future work
Slide 12
Greedy Heuristics
 Greedy RTB
 Greedy RBT
 Greedy RR
 Greedy CLR
 Greedy CRL
 Greedy CR
Slide 13
Overview
 The classical assignment problem
 Exact Solution methods
 Successive shortest path method
 A maximum matching algorithm
 Hungarian method
 Greedy heuristics
 Comparison
 Future work
Slide 14
Comparisons
Solution times of exact methods
 Benchmark set 1: JE Beasly (Randomly Generated)
60.0 1024 MB ram, Windows XP
50
 3.4 Ghz,
time (seconds)
40
35
30
40.0
30.0
25
20
20.0
15
10
10.0
5
0
0.0
number of iterations
45
50.0
100
200
300
400
500
600
700
800
Hungarian
Hungarian
0.5
0.4
1.0
1.9
3.1
3.9
5.1
6.1
MWM
MWM
1.6
8.9
10.9 37.5 36.4 9.9
MWM Iterations
44
38
14
15
7
size
Slide 15
2
49.3 23.4
4
3
MWM Iterations
Expon. (MWM)
Comparisons
Solution times
6
RTB
time (seconds)
5
RBT
RR
4
CLR
CRL
3
CR
2
1
0
100
200
300
400
500
size
Slide 16
600
700
800
Comparisons
% away from optimal
1.2
% optimal
1
RTB
RBT
0.8
RR
0.6
CLR
CRL
0.4
CR
0.2
0
100
200
300
400
500
size
Slide 17
600
700
800
Comparisons
Solution times of exact methods
Benchmarks set 2: Randomly Generated in Matlab
1200
800
600
Hungarian
400
MWM
200
SSP
60
70
80
90
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
90
10 0
00
0
10
20
30
40
50
time (seconds)
1000
size
Slide 18
Expon.
(SSP)
Comparisons
Solution time
300
RTB
200
RBT
RR
150
CLR
CRL
100
CR
50
size
Slide 19
80
0
10
00
30
00
60
0
40
0
20
0
90
70
50
30
0
10
time (seconds)
250
Comparisons
% away from optimal
4
3.5
RTB
RBT
2.5
RR
2
CLR
1.5
CRL
CR
1
0.5
size
Slide 20
30
00
10
00
80
0
60
0
40
0
20
0
90
70
50
30
0
10
% optimal
3
Future work
 Advanced Heuristics and Meta-heuristics
 More exact solution methods
 Expand algorithms to solve variations of the
assignment problem
Slide 21
References
[1]
[2]
[3]
[4]
[5]
Slide 22