solution methodologies for the classical grant van dieman friday 30 th november 2007 supervisor:...
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Solution methodologies for the classical
Grant van Dieman
Friday 30th November 2007
Supervisor: Prof. JH van VuurenCo-supervisor: Mr JN Roux
assignment problem
Slide 2
Overview The classical assignment problem
Exact Solution methods A maximum matching algorithm Successive shortest path method Hungarian method
Greedy heuristics
Comparison
Future work
Slide 3
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 4
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 5
Overview The classical assignment problem
Exact Solution methods A maximum matching algorithm Successive shortest path method Hungarian method
Greedy heuristics
Comparison
Future work
Slide 6
A maximum matching algorithm for weighted bipartite graphs (MWM)
qij
V1 = {assignees}
V2 = {tasks}
G :
Slide 7
A maximum matching algorithm for weighted bipartite graphs (MWM)
V1 = {assignees}
V2 = {tasks}
qij M :
Slide 8
Overview The classical assignment problem
Exact Solution methods A maximum matching algorithm Successive shortest path method Hungarian method
Greedy heuristics
Comparison
Future work
Slide 9
Successive shortest path algorithm(SSP)
Minimum cost flow algorithm
Why this algorithm can be used to solve the assignment problem
The value of xij will be binary
.,
,,
, subject to
Minimize
,:,
Ejilx
Ejiux
Viibxx
xc
ijij
ijij
Ejijij
Eijj:ji
ijEi,jij
Slide 10
Overview The classical assignment problem
Exact Solution methods Successive shortest path method A maximum matching algorithm Hungarian method
Greedy heuristics
Comparison
Future work
Slide 11
Hungarian Method
Kuhn(1955)
Special algorithm for the assignment problem
Construct reduced cost matrix
Slide 12
Overview The classical assignment problem
Exact Solution methods Successive shortest path method A maximum matching algorithm Hungarian method
Greedy heuristics
Comparison
Future work
Slide 13
Greedy Heuristics
Greedy RTBGreedy RBTGreedy RR
Greedy CLRGreedy CRLGreedy CR
Slide 14
Overview The classical assignment problem
Exact Solution methods Successive shortest path method A maximum matching algorithm Hungarian method
Greedy heuristics
Comparison
Future work
Slide 15
Comparisons
Benchmark set 1: JE Beasly (Randomly Generated) 3.4 Ghz, 1024 MB ram, Windows XP
Slide 16
Comparisons
Solution times
0
1
2
3
4
5
6
100 200 300 400 500 600 700 800
size
tim
e (s
econ
ds)
RTB
RBT
RR
CLR
CRL
CR
Slide 17
Comparisons
% away from optimal
0
0.2
0.4
0.6
0.8
1
1.2
100 200 300 400 500 600 700 800
size
% o
pti
mal
RTB
RBT
RR
CLR
CRL
CR
Slide 18
Comparisons
Benchmarks set 2: Randomly Generated in Matlab
Slide 19
ComparisonsSolution time
0
50
100
150
200
250
300
10 30 50 70 90 200
400
600
800
1000
3000
size
tim
e (s
econ
ds)
RTB
RBT
RR
CLR
CRL
CR
Slide 20
Comparisons
% away from optimal
0
0.5
1
1.5
2
2.5
3
3.5
4
10 30 50 70 90200 400 600 800
1000
3000
size
% o
ptim
al
RTB
RBT
RR
CLR
CRL
CR
Slide 21
Future work
Advanced Heuristics and Meta-heuristics
More exact solution methods
Expand algorithms to solve variations of the assignment problem
Slide 22
References
[1]
[2]
[3]
[4]
[5]