solution methodologies for the classical grant van dieman friday 30 th november 2007 supervisor:...

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]