A Neural Network Approach

8/4/2019 A Neural Network Approach

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EUROPEANJOURNALOF OPERATIONALRESEARCHE L S E V I E R Europ ean Journ al of Op erational Researc h 89 (1996) 556-563

T h e o r y a n d M e t h o d o l o g yA n e u r a l n e two r k ap p r oac h to fac il it y l ayou t p r ob l e m s

K a z u h i r o T s u c h i y a a ,b ,c ,, , S u n i l B h a r i t k a r c , Y o s h i y a s u T a k e f u j i b,ca Software a nd System Lab., F uji Electric Co. Research and Development, Ltd., I, Fuji-machi, Hino, To kyo I91, Japan

b Faculty of Enviro nm ental Information, Keio U niversity , 5322 Endoh, Fujisawa 252, Japanc Department of Electr ical Engineering and Applie d Physics, Case Western Reserve U niversity, Cleveland, O H 44106, U SA

Received 17 Jan uary 1994; revised 1 F ebrua ry 1995

Abstract

A near-opt imum paral le l a lgor i thm for solving faci l i ty layout problems is presented in th is paper where theproblem is NP-complete . The faci li ty layout problem is one o f the m os t fundam ental quadrat ic ass ignment problem sin Operations Research. The goal of the problem is to locate N facilities on an N-square (location) array so as tominimize the total cost. The proposed system is composed of N N neurons based on an artif icial two-dimensionalmaximum neu ral netw ork for an N-facility layou t problem. Ou r algo rithm has given improved solutions for severalbenchmark problems over the best existing algorithms.K e y w o r d s : Facility layout; Qu adra tic assignme nt problem; Neu ral networks; Two-dimensional maximum ne uronmodel ; Paral le l a lgor i thms

1 . I n t r o d u c t i o n

T h e f a c il i ty l a y o u t p r o b l e m i s o n e o f t h e t r a d i-t i o n a l q u a d r a t i c a s s i g n m e n t p r o b l e m s o r i g i n a l l yp r e s e n t e d b y K o o p m a n s a n d B e c k m a n n i n 1 95 7[ 13 ]. T h e p r o b l e m h a s b e e n w i d e l y s t u d i e d b ym a n y r e s e a r c h e rs i n O p e r a t i o n s R e s e a r c h a n dm a n a g e m e n t s c i e n c e , a n d k n o w n t o b e N P - c o m -p l e t e ( N P , n o n d e t e r m i n i s t i c p o l y n o m i a l ) [ 18 ].G o o d r e v i e w s h a v e b e e n s u m m a r i z e d i n [ 1 4 , 7 ]a n d s e v e r a l b e n c h m a r k p r o b l e m s h a v e b e e n g i v e nf o r c o m p a r i n g t h e a l g o r i t h m s [ 1 6 , 2 0 ] . G i l m o r e [ 6 ]a n d L a w l e r [ 1 5 ] i n d e p e n d e n t l y d e v e l o p e d b r a n c ha n d b o u n d a l g o r i t h m s to f i n d a n o p t i m u m s o lu -t i o n . S e v e r a l o t h e r b r a n c h a n d b o u n d a l g o r i t h m s

* Corresponding auth or. Fax: +81-425-86-8016.

h a v e b e e n p r o p o s e d b y P ie r c e a n d C r o w s t o n [ 17 ],B u r k a r d [ 4 ], a n d B a z a r a a [ 2 ]. A n o t h e r o p t i m u ma l g o r i t h m c a l l e d c u t t in g p l a n e a l g o r i t h m w a s p r e -s e n t e d b y B a z a r a a a n d S h e r a l i [ 3 ] a n d B u r k a r da n d B o n n i n g e r [ 5 ]. T h e o p t i m u m s o l u ti o n s h a v eb e e n c o n f i r m e d o n l y u p t o t h e 1 5 -f ac i li ty b e n c h -m a r k p r o b l e m . B e c a u s e o f t h e p r o h ib i t iv e l y l o n gc o m p u t a t i o n a l t i m e r e q u i r e d b y t h e o p t i m u m a l -g o r i th m s , t h e n e a r - o p t i m u m a l g o r i th m s h a v e b e e ns t u d i e d i n o r d e r t o g e n e r a t e a " g o o d " s o l u t i o n i na s h o r t t i m e . S o m e o f t h e e a r l i e r n e a r - o p t i ma l g o r i t h m s a r e H 6 3 [ 9 ] , H C 6 3 - 6 6 [1 0], C R A F T[1 1], F L A C [ 1 9 ], F R A T [1 2], a n d B i a s e d S a m p l i n g( B S ) [ 1 6 ] . R e l a t i v e ly n e w n e a r - o p t i m u m a l go -r i t h m s a r e b a s e d o n s i m u l a t e d a n n e a l i n g [2 4,8 ] o rt a b u s e a r c h ( T A B U ) [ 2 0 ]. O n e o f t h e b e s t s im u -l a t e d a n n e a l i n g m e t h o d s f o r t h e f a c il it y l a y o u tp r o b l e m i s H S A p r o p o s e d b y H e r a g u e t a l. [8 ],

0377-2217/96/$15.00 1996 Elsevier Science B.V. All rights reservedS S D I 0377-2217(95 )00051-8


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K. Tsuchiya et al. / European Journal of Operational Research 89 (1996) 556-563 55 7w h i c h o u t p e r f o r m e d a l l t h e e x i s t i n g s c h e m e s i nt e r m s o f s o l u t i o n q u a l i t y . H S A m u s t p r e p a r e a" g o o d " i n i t i a l s o l u t i o n a n d u s e t h e s i m u l a t e da n n e a l i n g t o o b t a i n a b e t t e r s o l u t i o n w h e r e s e v -e r a l p a r a m e t e r s a r e t o b e t u n e d .

T h e g o a l o f t h e p r o b l e m i s t o l o c a t e N f a c il i-t i e s o n a n N - s q u a r e ( l o c a t i o n ) a r r a y s o a s t om i n i m i z e t h e t o t a l c o s t . T h e m a t h e m a t i c a l e x -p r e s s i o n o f t h e N - f a c i l i t y l a y o u t p r o b l e m i s g i v e nby :

M i n i m i z e t h e t o t a l c os t:N N N NE E E E ci,jdp,qVp,qVi,pl/5,q ( 1 )i = l j - i + l p - I q = l

s u b j e c t t oN

E V/,p = 1, fo r p = 1 . .. . N , (2 )i= 1

NY'~ Vi,p = l , f o r i = l ..... N , ( 3 )p - I

10 i f f ac i l i ty # i i s a s s ig ned toV~,p = loca t ion # p , (4 )

o t h e r w i s e ,w h e r e c i , j i s t h e c o s t b e t w e e n f a c il i ty # i a n df a c i l i t y # j , d p , q i s t h e m a n h a t t a n g r i d d i s t a n c eb e t w e e n l o c a t i o n # p a n d l o c a t i o n # q , c i , j = c j , i ,a n d d p , q = d q , p . N o t e t h a t t h e c o s t m e a n s t h ef l o w o r t h e w e i g h t b e t w e e n t w o f a c i l it i e s . F i g . l ( a )a n d F i g . l ( b ) s h o w a s o l u t i o n o f t h e 5 - f a c i l i tyl a y ou t p r o b l e m a n d t h e c o s t / d i s t a n c e t a b le r e -s p e c t i v e l y w h e r e 5 f a c i li t i e s a r e a l l o c a t e d o n a5 - s q u a r e a r r a y . T h e t o t a l c o s t i s 3 3 ( = C l , 2 d l , 2 q -Cl ,3d l , 3 + c " ( 1 ,4d l ,4 + Cl ,5d l , 5 --I-c2,3d2 ,3 + c2,4 d2, 4 -- l-c2, 5


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5 5 8 K. T suchiya e t a l . / Euro pean Journal o f Operat ional Research 89 (1996) 556- 563I n t h i s p a p e r , a t w o - d i m e n s i o n a l m a x i m u m n e u r a ln e t w o r k is n e w l y i n t r o d u c e d f o r s o l v in g t h e f a ci l-i ty l a y o u t p r o b l e m , a n d t h e q u a d r a t i c a s s i g n m e n tp r o b l e m i n g e n e r a l .

I n t h e n e x t s e c t io n , w e r e v i e w t h e b a s ic m e c h a -n i s m o f th e a r ti f ic i al n e u r a l n e t w o r k f o r o p t i m i z a -t i o n a n d c o m b i n a t o r i c s , a n d e x p l a i n o u r n e u r a ln e t w o r k r e p r e s e n t a t i o n f o r t h e f a c il it y l a y o u tp r o b l e m . I n S e c t i o n 3 , w e d e s c r i b e t h e n e u r a ln e t w o r k p a r a l l e l a l g o r i t h m , a n d t h e n d i s c u s s t h ee x p e r i m e n t a l r e s u l t s i n S e c t i o n 4 w h e r e s e v e r a lb e n c h m a r k p r o b l e m s a r e u s e d t o s h o w t h e e f f e c -t i v e n es s o f o u r a l g o r i th m . W e s u m m a r i z e t h isp a p e r i n S e c t i o n 5 .

2 . T h e n e u r a l n e t w o r k r e p r e s e n t a t i o nT h e m a t h e m a t i c a l m o d e l o f t h e a r t if i ci a l n e u -

r a l n e t w o r k c o n s is t s o f t w o c o m p o n e n t s ; n e u r o n sa n d w e i g h t e d s y n a p t i c l i n k s . A n o u t p u t s i g n a lt r a n s m i t t e d f r o m a n e u r o n p r o p a g a t e s t o o t h e rn e u r o n s a s a n i n p u t s i g n a l t h r o u g h t h e s y n a p t i cl in k s . E v e r y a rt if i ci a l n e u r o n h a s t h e i n p u t U a n dt h e o u t p u t V . T h e i n p u t o f th e i , j - t h n e u r o n U~dis u p d a t e d b y a m o t i o n e q u a t i o n w h i ch r e p r e s e n t st h e w e i g h t e d s y n a p t i c l i n k s b e t w e e n t h e i , j - t hn e u r o n a n d o t h e r n e u r o n s . T h e o u t p u t o f t h ei , j - t h n e u r o n i s g i v e n b y V id = f ( U / d ) w h e r e f isc a ll e d t h e n e u r o n ' s i n p u t / o u t p u t f u n c t i o n . N o t et h a t V /d i s a l s o u s e d i n E q . 1 . T h e o u t p u t o f t h ei , j - t h n e u r o n V /d = 1 m e a n s t h a t f a c i l it y # i i sa s s i g n e d t o l o c a t i o n # j .

T h e i n p u t / o u t p u t f u n c ti o n o f t h e t w o -d i m e n -s i o n a l m a x i m u m n e u r o n i s g i v e n b y :1 . V a ,b = 1 i f U a , b = m a x { U / , i } ;2 . V c ,d = 1 i f U c , d = m a x { U i , / l i # a , j ~ b } ;3 . V e , f = 1 i f U e , f = m a x { U i , i [ i ~ a , c , j ~ b , d } ;N . V g ,h = 1 i f U g , h = m a x { U / d l i ~ a , c , c . . . . .

j ~ b , d , f . . . . };V / ,j = 0 o t h e r w i s e , w h e r e i 4~ a , c , e , . . . , g ,

j 4 : b , d , f , . . . , h .(5 )

F o r e x a m p l e , f a c il it y # a s h o u l d b e a s s i g n e d t o

l o c a t i o n # b i f U a ,b i s t he l a rge s t a mong a l l U~ d ' s ,a n d t h e n f a c il it y # c s h o u l d b e a s s i g n e d t o lo c a -t i o n # d i f U c ,d i s t he l a rge s t a mong a l l U~ , / se x c e p t f o r i = a o r j = b . N o t e t h a t i f t w o o r m o r et h a n t w o n e u r o n s h a v e t h e s a m e l a r g e s t i n p u t ,o n e n e u r o n s h o u l d b e s e l e c t e d a r b i t r a r e l y f r o ma m o n g t h e m . T h e s y s te m is c o m p o s e d o f N Nn e u r o n s . N n e u r o n s a lw a y s g e n e r a t e n o n z e r oo u t p u t s a n d t h e o t h e r ( N 2 - N ) n e u r o n s g e n er -a t e z e ro . T h i s t w o - d im e n s i o n a l m a x i m u m n e u r o nm o d e l s a ti s fi e s t h e c o n s t r a i n t s o f E q s . ( 2 ) - ( 4 ) .

T h e m o t i o n e q u a t i o n o f t h e i , j - t h n e u r o n i sg e n e r a l l y g i v e n b y :dU/ d c ': qE V l , 1 . . . . , V i, j . . . . . V N , N )

d t = OV/d , (6 )w h e r e E is t h e c o m p u t a t i o n a l e n e r g y f u n c t i o nf o l l o w i n g a n N N - v a r i a b l e f u n c t i o n : E ( V 1 ,1 . . . . .V i, . . . . . V N , N ) . T h e a r t i f i c i a l n e u r a l n e t w o r k p r o -v i d e s a g r a d i e n t d e s c e n t m e t h o d s o a s t o m i n i -m i z e t h e f a b r i c a t e d e n e r g y f u n c t i o n E . U s u a l lyt h e r i g h t t e r m i n E q . ( 6 ) c a n b e c o n s t r u c t e d b yc o n s i d e r i n g t h e n e c e s s a r y a n d s u f f i c ie n t c o n -s t r ai n t s a n d / o r t h e c o s t f u n c t i o n f r o m t h e g iv e np r o b l e m b e c a u s e t h e y g i v e i n f o r m a t i o n o n i n t e r -c o n n e c t i o n s o f t h e a r ti fi c ia l n e u r a l n e t w o r k . T h ec o m b i n a t i o n o f th o s e c o n s t r a i n t s a n d t h e c o s tf u n c t i o n , h o w e v e r , c o m p l i c a t e s t h e m o t i o n e q u a -t i o n a n d m a k e s i t es p e c i al l y d i f fi c u lt to c o m p u t et h e c o s t f u n c t i o n . T h e p r o p o s e d t w o - d i m e n s i o n a lm a x i m u m n e u r a l n e t w o r k n e e d s o n l y t h e c o s tf u n c t i o n s i n c e a l l t h e n e c e s s a r y a n d s u f f i c i e n tc o n s t r a in t s , n a m e l y E q s . ( 2 ) - ( 4 ) , a r e i n c l u d e d i nt h e n e u r o n m o d e l . I t p r o v i d e s a f a s t e r c o n v e r -g e n c e s p e e d a n d h i g h e r c o n v e r g e n c e ra t e t h a nt h o s e o f c o n v e n t i o n a l M c C u l l o c h - P i t t s n e u r o n o rs i g m o i d a l n e u r o n m o d e l s w h i c h r e q u i r e b o t h t h o s ec o n s t r a i n t s a n d t h e c o s t f u n c t i o n i n t h e m o t i o ne q u a t i o n .

T h e m o t i o n e q u a t i o n o f th e i , j - t h n e u r o n f o rt h e f a c i li t y l a y o u t p r o b l e m i s g i v e n b y :dU/ d = Q - R , ( 7 )d t

w h e r e Q a n d R a r e t h e o b je c t iv e c o s t a n d t h er e a l t o t a l c o s t r e s p e c t i v e l y . Q c a n b e s e t b y a u s e ra s a n e x p e c t e d t o t a l c o s t o r e v e n s e t t o b e z e r o .


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K. Tsuchiya et al. / European Journal of O perational Research 89 (1996) 556-563 55 9O u r s im u l a t io n i s t e r m i n a t e d w h e n R r e a c h e s Q .I t m e a n s t h a t d U ~ , J d t i s a l w a y s n e g a t i v e o r z e r o .R is g i v e n b y E q . ( 1 ) w h e r e i t i s a s s u m e d t h a tf a c il i t y # i is a l l o c a t e d t o l o c a t i o n # j . F o r e x a m -p l e , t o c a l c u l a t e d U 2 , 1 / d t a t t i m e t i n F i g . l ( a ) ,f a c il i ty # 2 i n l o c a t i o n # 2 i s a l l o c a t e d t o l o c a t i o n# 1 t e m p o r a r i l y a s s h o w n i n F ig . 3 w h i l e f a c i li t y# 1 i n l o c a t i o n # 1 is m o v e d t o l o c a t i o n # 2 t e m -p o r a r i l y . T h e r e a l t o t a l c o s t R i s 2 8 . I f Q = 2 0 ,t h e n d U 2 . 1 / d t = 2 0 - 2 8 = - 8 . T o c a lc u l a ted U 2 , 2 / d t , t h e a s s i g n m e n t o f fa c il it y # 2 t o l o c a-t i o n # 2 m u s t b e m a i n t a i n e d b e c a u s e f a ci li ty # 2h a s b e e n a c t u a l l y a s s i g n e d t o l o c a t i o n # 2 a t t i m et . S i n c e R i s 3 3 , d U 2 , 2 / d t = - 1 3 . E q . ( 7) d e -s c r ib e s t h e d e g r e e o f p e n a l t y w h i c h d i s c o u r a g e st h e h i gh l y p e n a l i z e d n e u r o n s f r o m g e n e r a t i n gn o n z e r o o u t p u t s .I n o r d e r t o i m p r o v e t h e g l ob a l m i n i m u m c o n -v e r g e n c e a n d t o a c c e l e r a te t h e s i m u l a t i o n s p e e d[ 23 ], E q . ( 7 ) w a s r e p l a c e d b y :d Ui o = [ ( Q - R ) V i j i f ( t m o d l 0 ) < t o ,

d t ~ Q - R o t h e r w i s e , ( 8 )w h e r e t a n d t o a r e th e n u m b e r o f i t e r a ti o n s t e p sa n d a c o n s t a n t p a r a m e t e r r e s p e ct iv e l y . I n t h e f ir s te q u a t i o n , d U i , J d t = 0 i f V ~,j = 0 . I n o t h e r w o r d s ,t h e f i rs t e q u a t i o n is a c t i v a t ed a n d i m p o s e s t h ep e n a l t y o n t h e i , j - t h n e u r o n o n l y w h e n it g e n e r -a t e s a n o n z e r o o u t p u t . I t g i ve s z e r o - g e n e r a t i n gn e u r o n s c h a n c e s to g e n e r a t e n o n z e r o o u t p u t s a n da l l o w s t h e s y s t e m t o e s c a p e f r o m l o c a l m i n i m a .T h e to is o n l y o n e p a r a m e t e r a n d h e l p s t h e s t a t eo f t h e s y s t e m t o a v o i d s t a c k i n g a t t h e l o c a l r a in -

f a c i l i t y#1

f a c i l i t y#3

f a c i l i t y#2

f a c i l i t y#4 f a c i l i t y#5

Fig. 3. How to calculate dUe,1 / d t .

i m a. T h e c o n v e r g e n c e t h e o r e m w i t h p r o o f f o r t h et w o - d i m e n s i o n a l m a x i m u m n e u r a l n e t w o r k i sg i ve n in A p p e n d i x A .

3 . P a r a l l e l a l g o r i t h mA s i m u l a t o r b a s e d o n t h e p r o p o s e d n e u r a l

n e t w o r k w a s d e v e l o p e d a n d a s y n c h r o n o u s p a r a l -l el s y s t e m w a s s i m u l a t e d . T h e f i r s t- o r d e r E u l e rm e t h o d w a s u s e d t o s o lv e N 2 e q u a t i o n s i n E q . (8 )n u m e r i c a l ly . T h e f o l lo w i n g s t e p s d e s c r ib e t h ep r o p o s e d a l g o r i t h m f o r a n N - f a c il i ty la y o u t p r o b -l e m . N o t e t h a t t l im i t is t h e m a x i m u m n u m b e r o fi t e r at i o n s t e p s f o r t h e s y s t em t e r m i n a t i o n c o n d i -t i o n .

S t e p O . S e t t = 0 , a n d s e t t _ l i m i t , Q , a n d w f o rt h e N - f a c i li t y p r o b l e m .

S t e p 1 . I n i t i a l i z e v a l u e s o f U , j ( t ) f o r i , j =1 . . . , N u s i n g u n i f o r m r a n d o m n u m b e r s .

Table 1Summary of the total costsN-facil ity problem H63 [9] HC63- 66 [10] CR AF T [1] BS [16] FLA C [19] FR AT [12] TA BU [20] HS A [8] This work

5 [16] 25 29 25 25 25 28 - 25 256116] 43 43 43 43 43 45 - 43 437116] 77 74 74 74 74 80 - 74 748116 ] 109 107 107 107 107 111 - 107 10712116] 300 304 289 289 289 302 - 289 28915116] 617 578 583 575 585 602 575 575 5752011 6] 1384 1319 1324 1304 1303 1335 1285 1285 128530 [16] 3244 3161 3148 3093 3079 3160 3062 3062 306242[20] . . . . . . 7932 7927 792649[20] . . . . . . 11768 1 1 7 3 9 1 1 7 3 2-: not available.


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560 K. Tsuchiya et al. / European Journal of Operational Research 89 (1996) 556 -563S t e p 2 . E v a l u a t e V i , j ( t ) f o r i , j = 1 , . . . , N , u s i n g

E q . 5 .S t e p 3 . C o m p u t e E q . (8 ) f o r t h e N N n e u r a l

n e t w o r k f o r i , j = 1 . . . . N t o o b t a i n A U , .j (t ):dU~,j (9)A U / d ( t ) = a t

S t e p 4 . AU /d( t ) >__ O , th en ge ne ra t e the so lu t iona n d t e r m i n a t e t h i s p r o c e d u r e .

S t e p 5 . U p d a t e U / d (t + 1 ) f o r i , j = 1 . . . . . N ,b a s e d o n t h e f i r s t - o r d e r E u l e r m e t h o d :U , .d ( t + 1 ) = U id ( t ) + A U/ , j ( t ) . ( 10 )

S t e p 6 . E v alu a te V~d(t + 1 ) fo r i , j = 1 . . . . . N ,us ing E q . (5 ) .

S t e p 7 . I f t = t l i m i t, t h e n t e r m i n a t e t h i s p r o -c e d u r e ; e l s e i n c r e m e n t t b y 1 a n d g o t o S t e p 3.S t e p s 3 t h r o u g h 5 c a n r u n i n p a r a l l e l . T h e p r o -p o s e d a l g o r i t h m w a s te s t e d o n a S u n S p a r c l 0 a n da n H P 9 0 0 0 / 7 1 0 c o m p u t e r a l t h o u g h th e a l go -r i t h m i s e x e c u t a b l e e i t h e r o n a s e q u e n t i a l m a -c h i n e o r a p a r a l l e l o n e .

4 . E x p e r i m e n t a l r e s u l t s

8 6 29 34 11 42 253 38 40 15 7 17 411 37 21 9 12 32 5

16 24 4 20 26 19 1836 30 39 14 35 10 223 22 33 28 31 13 27

The t o t a l c o s t = 7926Fig. 4. An i mproved so lution for the 42-facility benc hmarkproblem.

22 23 3 48 12 40 49 22 23 3 48 12 40 4944 2 31 27 43 11 39 44 2 31 27 43 11 3937 16 6 38 45 29 46 37 33 6 38 45 20 4630 33 5 34 20 18 28 30 16 5 34 29 18 2821 8 10 35 41 14 25 21 8 10 35 41 42 2519 32 15 9 42 7 26 19 32 15 9 14 7 2617 4 24 1 13 36 47 17 4 24 1 13 36 47

T h e t o t a l cost = 11732 The total cost = 11732Fig. 5. Two different improved solutions for the 49-facilitybenchmark problem.

W e h a v e e x a m i n e d te n b e n c h m a r k p r o b l e m s t ot e s t o u r a l g o r i t h m . T a b l e 1 s u m m a r i z e s t h e s o l u -t i o n q u a l it y o f t h e p r o p o s e d a l g o r i t h m a n d t h o s eo f t h e b e s t e x i s t i n g a l g o r i t h m s . T h e p r o p o s e da l g o r i t h m g e n e r a t e d b e t t e r s o l u t i o n s i n e v e r yb e n c h m a r k p r o b l e m . O u r s i m u l a t o r d i s c o v e r e di m p r o v e d s o l u t i o n s i n t h e 4 2 - f a c il i ty a n d t h e 4 9 -f a c i l it y b e n c h m a r k p r o b l e m s . F i g . 4 s h o w s t h ec o n f i g u r a t i o n o f t h e 4 2 - fa c il it y p r o b l e m w i t h t h et o t a l c o s t o f 7 9 2 6 w h i l e t h e b e s t k n o w n t o t a l c o s twa s 7927 [8 ] . F ig . 5 (a ) and F ig . 5 (b ) show d i f f e r -e n t c o n f i g u r a t i o n s o f t h e 4 9 - f a c il i ty p r o b l e m w i t ht h e t o t a l c o s t o f 1 1 7 3 2 w h i l e t h e b e s t k n o w n t o t a lc o s t w a s 1 1 7 39 [ 8 ] . T h e s y s t e m c o n v e r g e d t o t h e s es o l u t i o n s w i t h i n 9 0 0 0 i t e r a t i o n s t e p s . T a b l e 2 d e -p i c ts a s u m m a r y o f t h e c o m p u t a t i o n t i m e f o r t h ep r o p o s e d a l g o r i t h m . T h e v a l u e o f t o u s e d f o re a c h b e n c h m a r k p r o b l e m i s a l s o s h o w n i n T a b l e2 . W h e n t h e p r o b l e m s i z e i s i n c r e a s e d , m o r ec o m p u t a t i o n t i m e i s r e q u i r e d . F i g . 6 s h o w s t h er e l a t i o n s h i p b e t w e e n t h e t o t a l c o s t s o b t a i n e d b y

t h e p r o p o s e d a l g o r i t h m a n d t h e c o n v e r g e n c e f i e -q u e n c y f o r t h e 3 0 - f a ci l it y l a y o u t p r o b l e m . N o t et h a t t h e r e s u l t w a s o b t a i n e d b y 1 0 0 0 s i m u l a t i o nr u n s u s i n g d i f f e r e n t in i ti a l u n i f o r m r a n d o m i n p u ts e t s . B e t t e r t o t a l c o s t s w e r e o b t a i n e d a s t l i m i tb e c a m e l a r g e r . F i g . 6 s h o w s t h a t o u r a l g o ' ~ i t h m

Table 2Summary of the computation time and value of toN-facility prob lem CPU time (s) to

5 [16] 0.04 96 [16] 0.04 97 [161 0.04 98 [161 0.04 7

12 [16] 0.06 515 [16] 0.34 520 [16] 2.64 330 [16] 56.01 342 [20] 1746 249 [20] 3334 2CPU times were measured on a HP 9000/710.


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K. Tsuchiya et al . / European Journal Of Operational Research 89 (1996) 556-563 561

o

12 010 0

80604020

min. total cost = 3062max . total cost = 3248av. total cos t = 3124std. dev. = 33.83f requency of convergenceto min. total co st = 8av. iterations = 1806t_l imi t= 3000Q = 3062~ = 3

03050 3100 3150 3200 3250

total costFig. 6. The d istribution of the total co sts for the 30-facility problem .

f o u n d " g o o d " s o l u t io n s e v e n f r o m r a n d o m i n it ia lc o n f i g u r a t i o n s .

A p p en d ix A . C on vergen ce p rop erty o f th e tw o-d i -m e n s i o n a l m a x i m u m n e u r a l n e t w o r k

5 . C on c lu s ionI n t h i s p a p e r w e h a v e p r o p o s e d a n e a r - o p t i -

m u m p a r a l le l a l g o r it h m b a s e d o n t h e t w o - d i m e n -s i o na l m a x i m u m n e u r a l n e t w o r k f o r fa c il it y l a y o u tp r o b l e m s . T h e p r o p o s e d a l g o r i t h m u s e s N Nn e u r o n s f o r a n N - f a ci li t y l a y o u t p r o b l e m , w h e r eo n l y o n e s i m p l e p a r a m e t e r to is n e e d e d t o t u n ea n d " g o o d " i n i t i a l c o n f i g u r a t i o n s a r e n o t r e -q u i r ed . T h e s i m u l a t i o n re s u l ts d e m o n s t r a t e d t h a to u r a l g o r i t h m i s c a p a b l e o f g e n e r a t i n g b e t t e r s o -l u t i o n s o v e r t h e e x i s t i n g a l g o r i t h m s f o r s o m e o ft h e m o s t w i d el y u se d b e n c h m a r k p r o b l e m s . N e w l yd i s c o v e r e d s o l u t i o n s a r e a b l e t o s u b s t a n t i a t e t h ee f f e ct i v e n e s s o f t h e p r o p o s e d a l g o r it h m .

W e a r e g o i n g t o n o t o n l y s o l v e l a r g e r s i z ep r o b l e m s b u t a l s o a p p l y t h e p r o p o s e d a l g o r i t h mt o o t h e r q u a d r a t i c a s s i g n m e n t p r o b l e m s i n t h ef u t u r e .

T h e c o n v e r g e n c e p r o p e r t y o f th e t w o - d i m e n -s i o n al m a x i m u m n e u r a l n e t w o r k i s d e t e r m i n e d b yt h e t i m e d e r i v a t i v e s o f t h e e n e r g y o f t h e s y s t e m ,d E / d t . L e m m a 1 i s i n t r o d u c e d t o p r o v e t h a t t h ep r o p o s e d s y s t e m is a lw a y s a ll o w e d t o c o n v e r g e t ot h e e q u i l i b r i u m s t a t e o r th e o p t i m a l ( n e a r - o p t i -m u m ) s o l u t i o n .I k m m a 1. d E / d t < 0 is s a t is f ie d u n d e r t h e f o l l o w -i n g t w o c o n d i t i o n s :

(1 ) d U ~ , i / d t = - b E / O V i d = Q - R a n d(2 ) T h e i n p u t / o u t p u t f u n c ti o n o f th e n e u r o n

m o d e l is g i v e n b y :1. Va.b = 12 . ~ , a = 13 . Ve , f = 1

if Ua ,b = m a x { U / j } ;i f Uc ,d = m a x{ U / j l i a , j * b} ;if Ue, f= m a x { U / ,j I i ~ a , c , j ~ b , d } ;

A c k n o w l e d g e m e n t sT h i s w o r k i s p a r t l y s u p p o r t e d b y F u ji E l e c t r i c

C o ., L td . a n d T E P C O S c ie n ce F o u n d a t i o n . T h ea u t h o r s a r e a l s o t h a n k f u l t o D r . J . S k o r i n - K a p o vf o r p r o v i d in g d a t a f o r t h e l a r g e r s i ze p r o b l e m s .

N . V g , h 1v , . j = o

i f U g ,h = m a x { U i j l i 4 : a , c , e . . . . .j 4 : b , d , f . . . . } ;

o t h e r w i s e .( A . 1 )


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562 K. Tsuchiya e t al. / European Journal o f Op erat ional Research 89 (1996) 556-563P r o o f . C o n s i d e r t h e d e r i v a t iv e s o f t h e c o m p u t a -t i o n a l e n e r g y E w i t h r e s p e c t t o t i m e t .d E d U / , j d V / , j 0 Ed t = ~ ] ~ d t d U / , j O l ~ / , ji j

( d U / , y / 2 d V / , j= - ~ . ~ . ~ d t ] d U / , 'j

w h e r e O E / O V i , j i s r e p l a c e d b y - d U i , j / d t ( C o n d i -t i o n 1 ). L e tdU i , j U i , j ( t + d t ) - U / , j ( t)

d t d t 'd V / , j V i , y ( t + a t ) - V / , y ( / )d U / 3 U , ., j( t + d t ) - U / 3 ( t ) "

A s s u m e t h a t th e i n p u t o f t h e ( a , b ) - th n e u r o n i so n l y c h a n g e d d u r i n g t i m e t a n d t + d t i n t h es y s t e m .

( d U / , j ) 2 d V / , j- l ~ j~ - , d t d U / 3

V / , y ( t + d t ) - V / , y ( t ) Ui , j ( t + d t ) - Ui,y( t )

= - E E U ~ j ( t + d t ) - U ~ , j( t )i j ( d t ) 2

( V ~ j ( t + d t ) - V / a ( t ) )U a , b ( t + d t ) - - U a , b ( t )

( d t ) 2

( V a , b ( t + d t ) - V o , b ( t ) ) .I t is n e c e s s a r y a n d s u f f ic i e n t t o c o n s i d e r t h ef o l lo w i n g t w o c a s e s ( C o n d i t i o n s 1 a n d 2 ):( 1 ) V ~ ,b( t + d t ) = V ~ , b ( t ) ;( 2 ) V a , b ( t + d t ) = 0 , V a , b ( t ) = 1 .I f C a s e 1 . is s a t is f i e d , t h e n

[ d U i y l 2 d V i j

b e c a u s e V ~ ,b (t + d t ) - V a , b ( t ) = O .

I f C a s e 2 is s a ti s f ie d , t h e n

b e c a u s e V a , b ( t + d t ) - V ~ , b ( t ) = - - 1 a n dU i d ( t + d t ) - U / , y ( t ) d U /, y

d t d t( C o n d i t i o n 1 ) .

T h e r e f o r e d E / d t


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K. Tsuchiya et al. / European Journal of Operational Research 89 (1996) 556-563 56 3[13 ] Ko o p m a n s , T .C . , a n d B e c k m a n , M. , "Ass ig n m e n t p ro b -lems and the loca t ion of economic ac t iv it ies" , Economet-rica 25 (1957) 53-76.[14] Kusiak , A. , and Heragu , S .S . , "The fac i l i ty layout prob-l e m " , European Journal of Operational Research 29 (1987)229- 251 .[15] Lawler , E.L., "T he quadra t ic assignm ent problem ",Management Science 13 (1963) 42-57.[16] Nugent , C.E., V ol lma nn, T.E. , and Rum l, J . , "A n experi -m e n ta l c o m p a r i so n o f t e c h n iq u e s fo r t h e a ss ig n m e n t o ffacili t ies to locations", Operations Research 16 (1968)150-173.[17] Pie rce , J .F . , and Crowston , W.B. , "T ree-search a lgo-r i thms for quadra t ic assignment problems", Naval Re-search Logistics Quarterly 18 (1971) 1-36.[18] Sahni , S . , and Gonza lez , T. , "P-comple te approximat ionp ro b le m " , Journal of Associated Computing Machinery23/3 (1976) 555-565.[19] Scr iab in , M., and Vegin , R.C. , "A c luste r ana ly t ic ap-

proach to facili ty layout", The Journal of Industrial Engi-neering 18/2 (1985) 690-695.[20] Skorin-Kapov, J . , "Ta bu search appl ied to the qu adra t ica ss ig n m e n t p ro b l e m " , OSRA Journal on Computing 2 / 1(1990) 33-45.[21] Take fuji , Y., and Lee, K.C., " A n artificial hysteresisb inary neuron: A model suppressing the osc i l la tory be-h a v io u r o f n e u ra l d y n a m ic s" , Biological Cybernetics 64(1991) 353-356.[22] Takefu j i , Y. , and Lee , K.C. , "A n a r t i f ic ia l maxim umn e u ra l n e two rk : A win n e r - t a k e -al l n e u ro n m o d e l fo rc in gthe s ta te of the system in a so lu t ion domain", BiologicalCybernetics 67 (1992) 243.[23] Takefuji , Y., Neural Network Parallel Computing, Klu we rAcad emic Publ ishers , Boston , 1992.[24] W ilhe lm, M.R., and W ard , T.L. , "Solv ing quadra t ic as-s ig n m e n t p ro b l em s b y s im u la t e d a n n e a l in g " , liE Trans-actions (1987) 107-119.

A Neural Network Approach

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