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(e)istered o&&ice: Morti'er ouse -/1 Morti'er 3treet London +1T J U4

International Journal ofProduction ResearchPublication details, including instructions for authors

and subscription information:

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A coevolutionary algorithm for a

facility layout problem

T Dunker

a

, G adons

b

! " #estk$mper

a

a %raunhofer &nstitute of 'anufacturing "ngineering and(utomation )obelstrasse *2 +0- tuttgart German

b T1 hemnit3 &nstitut f4r Phsik eichenhainer trasse+0 0*2- hemnit3 GermanPublished online: 0- (ug 20*0.

To cite this article: T Dunker , G adons ! " #estk$mper 520067 ( coe8olutionar algorithm

for a facilit laout problem, &nternational 9ournal of Production esearch, *:*, 6+;

600, D.doi.org/*0.*0=0/0020+06*000**=*2

"L*!3* 3(OLL 6O+, $O( !(T%L*

Taylor # $rancis 'a7es every e&&ort to ensure the accuracy o& all the in&or'ation 8the

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2015

int. j. prod. res., 2003,vol. 41,no. 15, 34793500

A coevolutionary algorithm for a facility layout problem

T. DUNKER{*, G. RADONS{and E. WESTKAMPER{

T!" #a#$% #%$"$n&" a '($)(+&!(na% a##%(a' &( &$ n+-$%!'a(#&!-!a&!(n (/ a%$ /a'!!& a(+&". O+% (% !" a"$d (n a -!$d!n&$$% -(d$ /(% &$ a(+& '(n"&%a!n&" and ($'&!)$", !'!-#%()$" /(%-+a&!(n" /(+nd !n &$ !&$%a&+%$. N$)$%&$$"", a(+&"!& -(%$ &an "$)$n d$#a%&-$n&" a%$ d!6'+& &( "()$. On$ a(+& !" &( a## $n$&!' a(%!&-""$a%'!n ""&$-a&!'a /(%"(+8&!(n" +& !&(+& +a%an&$$ (/ nd!n an (#&!-+-. :n &!"#a#$% $ "+$"& "(-$ !-#%()$d -+&a&!(n and '%(""8()$%(#$%a&(%". ;$&, !& !n'%$a"!n n+-$% (/ d$#a%&-$n&" a"( $n$&!'a(%!&-" &a$ )$% (n. :n &!" 'a"$ $ #%(#("$ &( +"$ add!&!(na"&%+'&+%$" !)$n ?$n)!%(n-$n&@ +nd$%( an $)(+&!(n, &((. N+-$%!'a$#$%!-$n&" )$%!/ &!" '($)(+&!(na% a##%(a'.

1. Introduction

On$ "+#%($- !n /a'&(% #ann!n '(n"!"&" !n d$&$%-!n!n ((d('a&!(n" (/ a "$& (/ d$#a%&-$n&" >(% -an+/a'&+%!n '$" (n a #ana%"!&$. T!" &a" !" 'a$d &$ /a'!!& a(+& #%($-. T$ ($'&!)$"%!$B d$"'%!$d &$ (%d ?((d@ a%$ -an!/(d. :n add!&!(n, -an (/&$- a%$ (/ a 199F#%$"$n& a d$&a!$d %$)!$ (/ &$ d!$%$n& /(%-+a&!(n" (/ &$ /a'!!& a(+&

R$)!"!(n %$'$!)$d Ma%' 2003.

{ %a+n(/$% :n"&!&+&$ (/ Man+/a'&+%!n En!n$$%!n and A+&(-a&!(n,N($"&%a""$ 12,705F9, S&+&&a%&, G$%-an.

{ TU =$-n!&, :n"&!&+& /+% P"!, R$!'$na!n$% S&%a""$ 70, 0912F =$-n!&, G$%-an. * T( (-'(%%$"#(nd$n'$ "(+d $ add%$""$d. $-a! &-dH!#a./.d$

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8oevolutionary algorithm for a facility layout problem '&*1

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>"? If i " Ifand P

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In order to describe the transformation of a fi


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8oevolutionary algorithm for a facility layout problem 34I3

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'&** . 3un2er et al.

'.1. An adapted genetic algorithm

In Rajase2haran et al. >1*? a >!#1? seGuence for setting the binary variables isused as the genetic code. Standard cross=over and mutation operators are applied.4bviously, some of the outcomes are infeasible settings. If an infeasible gene occurs

it is eliminated from the population immediately. here are for e


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8oevolutionary algorithm for a facility layout problem '&1

geneticValgorithm

initialie population and compute the average of the objectivevalues do

1 crfor 17 . . . 7 n_1select parents from generation

cross=over generates to ne individuals for generation

endfor

for 17 . . . 7 nmu

_1select parent from generationmutation generates a ne individual for

generation endforcopy the n

co best individuals from generation _1 to generation

compute the average of the objective values and determine

hether the best individual has changed

hile the change of the average is larger than mch

and the best

individual has changed during the last nnc

generations and _mge

Applying this to the third e1'? ith eight departments e obtained

satisfactory results. Running the deterministic algorithm >'1 h '! min on a Pentium III *))

5U ith a memory use of approon a

Pentium II &!! 5U?. Also for all other e


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'&" . 3un2er et al.

6

average: 8937.6

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deviation: 102

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frequency

3

2

1

0

8800 8900 9000 9100

best objective value

igure &. 3istribution of the objective values.

the best 2non from 3as >1'? and Rajase2haran et al. >1*? shoing aconsider=able improvement in all cases.

'.". 3escription of the coevolutionary algorithm

If e attac2 large problems ith the above=described simple genetic algorithm e

have to ait a very long time for good results. 4ne ay to obtain results more rapidly

could be stopping the algorithm if it e11? suggestedapproaching the facility layout problem in a hierarchical manner by a divide=and=conGuer strategy. hey formed groups, computed the layout for each of them,and placed the groups in a final step.

our=step method Oenetic algorithm Mest results of our umber of described in by Rajase2haran genetic algorithm asdepartments 3as >1'? et al. >1*? described above

* 1! $$$01 1$&0* * $$*0'1! 1% *$*0' 1 $$$0' 1% )&0%1" &1 ")$0% &% '%'0% '$ ')01

able ". 5inimal objective value for three e1'? hich arereported in 3as >1'? and Rajase2haran et al. >1*? in comparison ith our best results.


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8oevolutionary algorithm for a facility layout problem 3493

W$ #%(#("$ a '($)(+&!(na% -$&(d (/ !&$%a&!)$ na&+%$. :n a %"&"&$# $ nd !n!&!a a(+&" /(% $a' %(+#. N$&, $ & a %$'&an$a%(+nd $a' %(+# and $na%$ $a' "!d$ a /a'&(% 1 2 !)!n-(%$ "#a'$ &( $a' %(+# /(% #(""!$ /+%&$% 'an$. T!" a(", $..

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$na%$ &$ n$ %(+# a%$a

2endfor

hile &$%$ !" "&! a ?a%$@2/(% "(-$2 >(% &$ a)$%a$ (/ &$2@" !"a%$


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3494 . 3un2er $& a.

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&. Results and conclusions(% (+% n+-$%!'a $#$%!-$n&" $ '%$a&$d a %and(- $a-#$ !&

F2 d$#a%&8-$n&" >%$'&an$" (/ d!$%$n& "a#$". (% "!-#!'!& $

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&$ a##$nd! #%()!d$" a &$ n$'$""a% 2000 (n$ 'an nd %(+#!n a(%!&-" +"!n$n$&!' a(%!&-. On$ 'an '(n"&%+'& $a-#$" $%$ &$d$&$%-!n!"&!' $+%!"&!' Ja%aa!" et al. >1990 "&(#" /a% /%(- &$(#&!-+- a" !& 'an and$ (n "!-#$ $'an$ (#$%a&!(n". :n &$"$'a"$" a(%!&-" !$ &$ (n$ D$ !& et al. >2000 'an !-#%()$ &$%(+#!n.

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8oevolutionary algorithm for a facility layout problem 3495


2015

convergence of 21 runs with 4 groups

elapsed time in seconds

9 average: 4533,0181

x1000000 deviation: 390,23915

8 best: 3979,55

worst: 5213,89 objective value

7average: 4,06205E6

value deviation: 131592,77082

best: 3,93936E6

bestobjective6 worst: 4,44988E6

5

4

0 1 2 3 4 5

elapsed time in seconds x1000


8elapsed time in seconds

average: 5390,635

x1000000

deviation: 1160,37707

best: 3397,35 objective value7 worst: 7170,13 average: 4,35529E6


value best: 4,23898E6

6 worst: 4,44855E6

bestobjective

5

0 1 2 3 4 5 6 7



8

elapsed time in seconds

average: 4966,666

x10

00000 deviation: 887,92038

7

best: 3783,38 objective value

worst: 7294,18 average: 4,27436E6deviation: 64162,99454

value

best: 4,18614E66 worst: 4,44648E6

bestobjective

5

4

0 1 2 3 4 5 6 7



8elapsed time in seconds

average: 5699,116

x1000000 deviation: 1306,47389

7

best: 3098,17objective value

worst: 7687,68average: 4,39307E6

value

deviation: 112706,55086

6 best: 4,18422E6

worst: 4,64994E6

bestobjective

5

40 1 2 3 4 5 6 7


Downloadedby[MonashUniversity igure %. 8omputational results for problem P)" ith four, siobtained on a P8, Pentium IL, 1.% OU?.

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Downl



2015

'&) . 3un2er et al.

convergence of 11 runs without grouping

4,6elapsed time in seconds

x1000000

average: 237530,45455

4,5 deviation: 72538,13254best: 103948 objective value

worst: 331484 average: 4,26083E6


value4,4 best: 4,18105E6

worst: 4,38149E6

objective

4,3

bes

t

4,2

0,0 0,5 1,0 1,5 2,0 2,5 3,0


igure ). Summary of computational results of the genetic algorithm ithout grouping forP)". Chile the values of the objective function are in the same range as the resultsobtained ith grouping, the computation time e


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igure $. Mest layouts for the last three e1'? found by theabove=described OA. or the e


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8oevolutionary algorithm for a facility layout problem '&

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D35

D34

D24D53

D9D55

D60

D20D59 D36

D14 D13 D61 D18

D51D25D56D32D11 D28D50

D3 D22 D26 D19D31 D48 D7 D21D57 D16

D52 D47 D43D12 D1

D30

D33D23D15

D37 D29D27D58 D41 D4

D49D44

D2 D45D17D38 D39

D42D40

D8D5

D6

D10D46

D62D54

Result, Iter. 9, Total Cost: 3939362.0 (4996 sec.)

D33 D54D40 D62

D27

D45 D23 D1 D12 D58D38 D52 D37D42D46

D4D41 D31D22

D51D35 D7D48

D3 D28

D39D30 D8 D21

D10 D13D6D50 D16 D60 D56 D53 D57 D11

D61 D24 D18 D43D5 D2 D29D17

D34D32 D36

D59

D55 D26 D20D44D47 D49 D15D9

D19D14 D25


D17D45

D10 D37D13 D39

D8D6

D19D36

D41D24D3D18

D25D40 D35

D32

D4 D46D2 D22D49 D20D56

D5D55D57 D28

D23D21

D12D30

D60 D29

D62 D31 D16 D15

D1D52D50 D27 D47

D59D61 D58

D26 D11D44D34 D42D53D54

D43

D51D38

D48 D9 D33

D7 D14


D15D46

D13 D42 D10

D14 D52D27D56 D36D28 D19

D34D47 D29 D11D40 D39

D45 D6D44

D24

D25 D49D31D57

D59D9 D16D30

D20 D55 D4D8 D32 D18

D26D50 D12D2

D3D23

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D38

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D58 D53

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D39

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D43D35

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D20 D60D15 D17

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D8

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62 departments, objective function: 4181054.0

igure *. he best layouts for four, si


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Referenc

esvar,.andCang,J.,"!!!,acilitylayoutoptimiationusingsimulationandgeneticalgorithms.InternationalJournalof

Production Research, '*, &')#*'.8han, Z. 8., and ansri, U., 1&, A study of genetic crossover operations on the facilities

layout problem. 8omputers and Industrial Qngineering, "), %'$#%%!.


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'%!! 8oevolutionary algorithm for a facility layout problem

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Rajase2haran, 5., Peters, M. A. and :ang, T., 199I, A $n$&!' a(%!&- /(% /a'!!& a(+& d$"!n !n B$!$-an+/a'&+%!n ""&$-". International Journal of Production Research,'), 95110.

Schnec2e, L. andLornberger, O., 1997, J%!d $n$&!' a(%!&-" /(% '(n"&%a!n$d #a'$8-$n& #%($-". IQQQransactions on Qvolutionary 8omputation, 1, 2FF277.

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:ASTED :n&$%na&!(na =(n/$%$n'$,:ASTEDA=TA P%$"", ##. F571.

A coevolutionary algorithm for a facility layout problem..doc

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