tabu search - exampleszebpe83/heuristic/lectures/ts_applications.pdf · heuristic algorithms for...

Heuristic Algorithms for Combinatorial Optimization Problems 1

Petru Eles, 2010

Tabu Search - Examples

Petru Eles

Department of Computer and Information Science (IDA)Linköpings universitet

http://www.ida.liu.se/~petel/

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Petru Eles, 2010

u Search Examples

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Heuristic Algorithms for Combinatorial Optimization Problems

Tabu Search

Tab

Hardware/Software Partitioning

Travelling Salesman

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ftware Partitioning

em:

f there exists a direct

esponding processes

edge:

ability for hardware

rocess.

ommunication between pro-

ardware and software

Inp

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Ou

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Tabu Search

TS Examples: Hardware/Sout:

The process graph: an abstract model of a syst

❚ Each node corresponds to a process.

❚ An edge connects two nodes if and only i

communication channel between the corr

❚ Weights are associated to each node and

- Node weights reflect the degree of suit

implementation of the corresponding p

- Edge weights measure the amount of c

cesses

tput:

Two subgraphs containing nodes assigned to h

respectively.

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ftware Partitioning

P11

P8

P9

P12

P10

Software

P14


Tabu Search

TS Examples: Hardware/So

P6

P4P1

P7

P5

P3

P2

Hardware

P13

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ftware Partitioning

measure of the computation

of operations in process i;

m inside process i;

re implementation;

We

W 2

KiC

KiU

KiP

KiSO


Tabu Search

TS Examples: Hardware/Soight assigned to nodes:

iN = M CL Ki

CL M U KiU M P Ki

P M SO KiSO×–×+×+×

L is equal to the RCL of process i, and thus is aload of that process;

= Nr_o pi

Nr_kind_o pi------------------------------- ; Ki

U is a measure of the uniformity

= Nr_o pi

L_pathi------------------ ; Ki

P is a measure of potential parallelis

=wop j

op j SPi∈∑Nr_o pi

----------------------------- ; KiSO captures suitability for softwa

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ing: Cost Function

2iN

----------

ts

Th

Re


Tabu Search

Hw/Sw Partition

e cost function:

Q1 W 1ijE

ij( ) cut∈∑× Q2

W 2ijE

ij( )∃∑

W 1iN

----------------------i( ) Hw∈∑

N H---------------------------------------× Q3

W 2iN

i( ) Hw∈∑

N H------------------------------

Wi( ) Sw∈∑

N S-------------------–

×–+

strictions:

H_costii H∈∑ Max

H≤

S_costii H∈∑ Max

S≤

W iN Lim1≥ i Hw∈⇒

W iN Lim1≤ i Sw∈⇒

amount of Hw-Sw comm.

Ratio com/cmp of Hw part.

Difference of average weigh

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Petru Eles, 2010

ves&Neighborhood

lution xnow is the set of

g a node from its current

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Tabu Search

Hw/Sw Partitioning: Mo

Moves:

❚ The neighborhood N(xnow) of a certain so

solutions which can be obtained by movin

partition to the other one.

Hw/Sw Partitioning: TS Algorithm

• Construct initial configuration xnow= (Hw0, Sw0)start:for each solution xk ∈ N(xnow) do

• Compute change of cost function ∆Ck = C(xk) - C(xnow) end forfor each ∆Ck < 0, in increasing order of ∆Ck do

if not tabu(xk) or tabu_aspirated(xk) thenxnow = xkgoto accept

end ifend forfor each solution xk ∈ N(xnow) do

• Compute ∆C′k = ∆Ck + penalty(xk)end forfor each ∆C′k in increasing order of ∆C′k do

if not tabu(xk) thenxnow = xkgoto accept

end ifend for• Generate xnow by performing the least tabu moveaccept:if iterations since previous best solution < Nr_w_b then

goto startend ifif restarts < Nr_r then

• Generate initial configuration xnow considering frequenciesgoto start

end ifreturn solution corresponding to the minimum cost function


Tabu Search

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ning: TS Algorithm

ssible) is tried.

d the best possible non-tabu

u state is executed.

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Tabu Search

Hw/Sw Partitio

First attempt:

❚ An improving non-tabu move (the best po

Second attempt:

❚ Frequency based penalties are applied an

move is tried;

Third attempt:

❚ The move which is closest to leave the tab

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ning: The Tabu-List

u-list. Their reverse is tabu.

lem and of the

with large tabu tenures.

tabu restriction: stronger

variable:

olution

iven interval.

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Tabu Search

Hw/Sw Partitio

The last τ moves performed are stored in the tab

τ = tabu tenure (length of the tabu list)

The tabu tenure depends on the size of the prob

neighborhood: large problem sizes are coupled

The tabu tenure depends on the strength of the

restrictions are coupled with smaller sizes.

Tabu tenures are tuned experimentally or can be

❚ too small tenures ⇒ cycling

❚ too large tenures ⇒ deterioration of the s

❚ Recommended values: 7 ÷ 25

Tabu tenures can be selected randomly from a g

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ng: Tabu Aspiration

ion produced is better than
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Tabu Search

Hw/Sw Partitioni

The tabu status of a move is ignored if the solut

the best obtained so far.

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Petru Eles, 2010

Long Term Memory

ns each node has spent in

for diversification:

on, which favors the transfer eir current partition.

y of occurrences of the node rtain threshold.

e started from an initial e encountered previously.

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Tabu Search

Hw/Sw Partitioning:

Long term memory stores the number of iteratio

the hardware partition. This information is used

1. Application of a penalty to the cost functiof nodes that have spent a long time in th

2. A move is forbidden (tabu) if the frequencin its current partition is smaller than a ce

3. If the system is frozen a new search can bconfiguration which is different from thos

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artitioning: Penalty

k spent in the Hw partition.

∈ Hw

∈ Sw

Coefficients experimentally set to:

❚ CH=0.4

❚ CS=0.15.

Th


Tabu Search

Hw/Sw Pe penalized cost function:

∆C'k ∆Ck

∆Cii

∑Nr_of_nodes------------------------------- pen k( )×+=

where

pen k( )

CH

Node_in_Hwk

N iter----------------------------------×–

CS 1Node_in_Hwk

N iter----------------------------------–

×–

=

❚ Node_in_Hwk : number of iterations node

❚ Niter : total number of iterations;

❚ Nr_of_nodes : total number of nodes;

if nodek

if nodek

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for Node Movement

ccurrences of the node in its

:

H

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Tabu Search

w/Sw Partitioning: Thresholds

A move is forbidden (tabu) if the frequency of o

current partition is smaller than the threshold:

Node_in_Hwk

N iter---------------------------------- T H>

1Node_in_Hwk

N iter----------------------------------–

T S>

The thresholds have been experimentally set to

❚ TH=0.2

❚ TS=0.4.

if nodek ∈ Hw

if nodek ∈ Sw

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perimental Results

arch partitioning

CPU time (s)(time with SA)

0.008(0.23)

0.04(1.27)

0.19(2.33)

30.5(769)


Tabu Search

Hw/Sw Partitioning: Some Ex

Parameters and CPU times for Tabu Se(SPARCstation 10)

numbers of nodes

τ Nr_w_b Nr_r

20 7 30 0

40 7 50 0

100 7 50 0

400 18 850 2

Variation of cost function for TS partitioning with 400 nodes

1.95e+06

2e+06

2.05e+06

2.1e+06

2.15e+06

2.2e+06

2.25e+06

2.3e+06

2.35e+06

2.4e+062.45e+06

0 500 1000 1500 2000 2500 3000

1.99e+06

1.992e+06

1.994e+06

1.996e+06

1.998e+06

2e+06

2.002e+06

2.004e+06

0 500 1000 1500 2500 3000

1.99e+06

1.992e+06

1.994e+06

1.996e+06

1.998e+06

2e+06

1920 1925 1930 1935 1940 1945 1950

2000Number of iterations

Cos

t fu

ncti

on v

alue

Number of iterations


Cos

t fu

ncti

on v

alue

Cos

t fu

ncti

on v

alue

optimum at iteration 1941



Embedded Systems for Real-Time Applications: Analysis and Synthesis

Tabu Search

16

September 2010

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perimental Results

h 100 nodes

100 120 140

timum at iteration 76

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Tabu Search


Variation of cost function for TS partitioning wit

38000

40000

42000

44000

46000

48000

50000

52000

54000

56000

0 20 40 60 80


Cos

t fu

ncti

on v

alue

op

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perimental Results

1000400

SA

TS

KL

ithmic)

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Tabu Search


Partitioning times with SA, TS, and KL

0.001

0.01

0.1

1

10

100

1000

10 10020 40

Number of graph nodes (logar

Exe

cuti

on t

ime

(s)

(log

arit

hmic

)

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Petru Eles, 2010

Salesman Problem

then return to the initial city; find the tour with minimum

graph where the nodes nt the distance between ered in traversing all cities.

A seacdis

In g

Finrepciti


Tabu Search

TS Examples: Travelling

alesman has to travel to a number of cities and h city has to be visited once. The objective is totance.

raph theoretical formulation:

d the shortest Hamiltonian circuit in a completeresent cities. The weights on the edges represees. The cost of the tour is the total distance cov

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TSP: Cost Function

ny tour can be represented

j.

are adjacent cities in the

that minimizes:

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Tabu Search

If the problem consists of n cities ci, i = 1, .., n, a

as a permutation of numbers 1 to n.

d(ci,cj) = d(cj,ci) is the distance between ci and c

Given a permutation π of the n cities, vi and vi+1

permutation. The permutation π has to be found

d vi vi 1+( , ) d vn v1( , )+

i 1=

n 1–

∑

The size of the solution space is (n-1)!/2

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ves&Neighborhood

se tours obtained by

nt set of k links, in a way that

the tour after two links have

imated in order to select the

y a sequence of such

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Tabu Search

TSP: Mo

k-neighborhood of a given tour is defined by tho

removing k links and replacing them by a differe

maintains feasibility.

For k = 2, there is only one way of reconnecting

been removed.

❚ Size of the neighborhood: n(n - 1) / 2

❚ As opposed to SA, all alternatives are est

appropriate move.

❚ Any tour can be obtained from any other b

moves.

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ves&Neighborhood

02

4

67


Tabu Search

TSP: Mo

5

3

1

Permutation:[0 2 4 6 7 5 3 1]

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ves&Neighborhood

02

4

67


Tabu Search

TSP: Mo

5

3

1

links (v3,v1), (v4,v6)are removed

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ves&Neighborhood

02

4

67


Tabu Search

TSP: Mo

5

3

1

Permutation:[0 2 4 3 5 7 6 1]

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ves&Neighborhood

in the permutation):

by the pair i, j so that i < j.

ed incrementally:

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Tabu Search

TSP: Mo

vi is the city in position i of the tour (ith position

remove (vi, vi+1) and (vj, vj+1)

connect vi to vj and vi+1 to vj+1

All 2-neighbors of a certain solution are defined

The change of the cost function can be comput

∆C = d(vi,vj) + d(vi+1,vj+1) - d(vi,vi+1) - d(vj,vj+1)

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Tabu-Classification

everal pairs of cities have

e tour before the swap.

utes)?

ecoming tabu?

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Qu


Tabu Search

TSP: Move Attributes for

We have performed a move as result of which s

swapped their position in the tour:

❚ Cities x and y are such a pair;

position(x) and position(y): positions in th

position(x) < position(y).

estions:

1. What information do we store (move attrib

2. Using this information, which moves are b

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Tabu-Classification

ur with city x and city y

pectively.

ur with city x occupying

.

n(x).


Tabu Search


1. Vector(x, y, position(x), position(y))

❚ To prevent any swap from resulting in a to

occupying position(x) and position(y) res

2. Vector(x, y, position(x), position(y))

❚ To prevent any swap from resulting in a to

position(x) or city y occupying position(y)

3. Vector(x, position(x))

❚ To prevent city x from returning to positio

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Tabu-Classification

n(y).


Tabu Search


4. City x

❚ To prevent city x from moving LEFT.

5. City x

❚ To prevent city x from moving.

6. Vector(y, position(y))

❚ To prevent city y from returning to positio

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Tabu-Classification


Tabu Search


7. City y

❚ To prevent city y from moving RIGHT.

8. City y

❚ To prevent city y from moving.

9. City x and y

❚ To prevent both cities from moving.

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Tabu-Classification

ts the smallest amount of

ts a large amount of moves).

iveness.


Tabu Search


❚ Condition 1 is the least restrictive (preven

moves).

❚ Condition 9 is the most restrictive (preven

❚ Conditions 3, 4, 5 have increasing restrict

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TSP: Tabu List Size

n be relatively small.

to be larger.

possibly good vicinity.

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Tabu Search

τ has to be experimentally tuned.

❚ For highly restrictive tabu conditions τ ca

❚ For less restrictive tabu conditions τ has

❚ τ too small ⇒ cycling

❚ τ too large ⇒ exploration driven away from

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TSP: Tabu List Size

ies/3

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Tabu Search

Tabu list size:

❚ for conditions 4 and 7: nr_cities/4 ÷ nr_cit

❚ for conditions 5, 8, 9: nr_cities/5

❚ for conditions 1, 2, 3, 6: ≈nr_cities

Best results for conditions 4 and 7

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Long Term Memory

es an edge is visited.

ng tour is generated

frequently.

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Tabu Search

TSP:

Long term memory maintains the number of tim

After a certain number of iterations a new starti

consisting of edges that have been visited less

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perimental Results

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Tabu Search

TSP: Some Ex

100 city problem; optimal solution: C = 21247.

❚ Best solution C = 21352 (21255 for SA)

❚ Time = 210 s (Sun4/75) - (1340 s for SA)

❚ Standard deviation over 10 trials: 30.3;

(randomly generated starting tour!)

❚ Average cost: 21372

57 city problem; optimal solution: C = 12955

❚ Optimal solution in 109 s (673 s for SA).

(Sequent Balance 8000)

tabu search - exampleszebpe83/heuristic/lectures/ts_applications.pdf · heuristic algorithms for...

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