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Z E S Z Y T Y N A U K O W E P O L I T E C H N I K I Ś L Ą S K IE J S e ria : A U T O M A T Y K A z .l 18

1996 N r kol. 1338

C z e s ła w S M U T N I C K I P o lite c h n ik a W ro c ła w s k a

S C H E D U L I N G O F F L E X I B L E F L O W L I N E S A N D W O R K C E N T R E S S u m m a r y . T h e p a p e r d e a ls w ith a flow lin e p r o d u c ti o n / s e r v i c e s y s te m . I t c o n s is ts o f t h e s e q u e n c e o f p ro c e s s in g c e n tr e s , e a c h c e n te r h a s a n u m b e r o f i d e n tic a l p a r a lle l m a c h in e s , a n d th e r e a r e in te r m e d ia te b u ffe rs b e tw e e n c e n te r s . A n im p r o v e m e n t a p p r o x im a t i o n a lg o r ith m for th e p ro b le m o f fin d in g th e s c h e d u le w i t h m in im u m m a k e s p a n is p r e s e n te d .

S Z E R E G O W A N I E E L A S T Y C Z N Y C H L I N I I P R Z E P Ł Y W O W Y C H I G N I A Z D P R O D U K C Y J N Y C H

S t r e s z c z e n i e . W p r a c y ro z w a ż a n y j e s t s y s te m p r o d u k c y jn y /o b s łu g i z linię, p rz e p ly w o w ę . L in ia t a s k ła d a się z c ię g u c e n tró w o b r ó b c z y c h , p r z y c z y m k a ż d e c e n t r u m p o s ia d a p e w n ę lic z b ę id e n ty c z n y c h m a s z y n , a ta k ż e b u fo ry o o g ra n ic z o n e j p o je m n o ś c i p o ś re d n ic z ę c e w p r z e k a z y w a n iu z le ce ń p o m ię d z y c e n t r a m i . D la te g o p r o b le m u , z k r y te r iu m m in im a liz a c ji m a k s y m a ln e g o te r m in u z a k o ń c z e n ia z a d a ń , p r o p o n o w a n y j e s t p e w ie n a lg o r y tm p o p ra w .

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

T h e p a p e r d e a ls c h iefly w ith a p r o b le m { F P ) d e riv e d f ro m a h y b r id c o m b in a tio n o f tw o c la s s ic s c h e d u lin g p r o b le m s , n a m e ly th e flow sh o p a n d p a ra lle l s h o p , a n d d e s c rib e d b rie fly a s fo llo w s. T h e r e is a s e t o f p a r ts a n d a s e t o f p ro c e s s in g c e n te r s e a c h o f w h ic h h a s a s e t o f p a r a lle l id e n tic a l m a c h in e s . A p a r t is a s s o c ia te d w ith a s e q u e n c e o f o p e r a tio n s p r o c e s s e d a t s u c c e s s iv e c e n te r s , a n d a ll p a r ts flow th r o u g h c e n te r s in t h e s a m e o r d e r. A t a c e n te r , a p a r t c a n b e p r o c e s s e d o n a n y m a c h in e . M o re o v e r th e r e a r e b u ffe rs w i t h l im i te d c a p a c ity t h a t m e d i a te in tr a n s f e r r in g p a r t s b e tw e e n m a c h in e s /c e n te r s . W e w a n t to fin d a s c h e d u le t h a t m in im iz e s t h e m a k e s p a n , o n e o f t h e m o s t fre q u e n tly ’ u s e d c r ite r io n .

A r c h i te c t u r e o f a u to m a te d m a n u f a c tu r in g s y s te m s u s u a lly d o c s n o t a llo w to fo rm i n t e r n a l q u e u e s o f p a r t s ( t r a n s p o r t e d o n p a lle ts o r in c o n ta in e r s ) , o r lim its t h e le n g th o f t h e s e q u e u e s d u e t o b u ffe r siz e. W ith re s p e c t to F P , t h e fo llo w in g p r o b le m s ( w ith an

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176 C . S m n tn ic k i

in c r e a s in g d e s c rip tio n c o m p le x ity ) c an b e c o n s id e re d : (U ) s y s te m h a s b u ffe rs w ith in f in ite c a p a c ity o r f in ite h o w e v e r la r g e e n o u g h , (W ) s y s te m w o rk s w ith o u t b u ffe rs , (B ) s y s te m w o rk s w ith b u ffe rs o f fin ite c a p a c ity . B u ffe rs c a n b e lo c a te d in s y s te m B e it h e r b e fo re m a c h in e s ( M ) o r b e f o re c e n te r s (C ). If th e c a p a c ity of a b u ffer is g r e a t e r t h a n o n e , w e s h o u ld a ls o c o n s id e r th e b u ffe r s e rv ic e ru le , e .g . a r b it r a r y o r d e r ( A ), F I F O ( F ) , L I F O (L ).

H e n c e , e a c h p r o b le m F P c a n b e c h a r a c te ris e d b y a d d itio n a l tr ip l e q |/3 |7 w h e r e a re fe rs t o t h e s y s te m ty p e , f) to t h e b u ffe r ty p e , 7 to th e b u ffe r s e rv ic e ru le .

F P c r e a te s t h e b a s ic m o d e l for a b r o a d cla ss o f p ro b le m s c a lle d in th e l i t e r a t u r e th e f le x ib le f l o w line s c h e d u lin g . T h e s e a r e p u r e p r o b le m F P o r p r o b le m s o b ta in e d d ir e c tly f ro m F P b y i n tr o d u c in g a few sp e c ific a d d itio n a l a s s u m p tio n s , e.g . so m e p a r t s c a n s k ip o n e o r m o r e m a c h in e s d u r in g th e r o u te , b u ffers h a v e in fin ite c a p a c ity , t h e r e is a t r a n s p o r t tim e b e tw e e n c e n te r s , e tc . A re a c h re v ie w o f in d u s tr ia l a p p lic a tio n s , a m o n g o t h e r s in c h e m ic a l b r a n c h e s , p o ly m e r a n d p e tr o le u m in d u s tr y , c o m p u te r s y s te m s , te le c o m m u n ic a ­ tio n n e tw o r k s , F M S , s p a c e s h ip p ro c e s s in g , e tc ., o n e can fin d in [4].

A lth o u g h F P h a v e q u ite s im p le f o r m u la tio n , it is tro u b le s o m e fro m t h e a lg o r ith m ic p o i n t o f v ie w . I ts A / 3- h a rd n e ss e s s e n tia lly r e s tr ic ts th e s e t o f a p p r o a c h e s w h ic h c a n b e a p p lie d to so lv e t h e p r o b le m . F ro m th e l it e r a t u r e o n e c a n fin d e x a c t a lg o r ith m s b a s e d on t h e b r a n c h - a n d - b o u n d ( B & B ) [8] a n d m ix e d - in te g e r p r o g r a m m in g ( M I P ) , a s w ell a s a v a r ie ty o f a p p r o x im a tio n m e th o d s , sec t h e re v ie w in [4]. S o m e o f th e s e p r o c e d u r e s h a v e b e e n d e s ig n e d for s p e c ia l c a s e s, e.g . for t h e s y s te m s t h a t h a v e o n ly tw o c e n te r s , t h e o n e s w h e r e o n ly o n e o f th e c e n te rs h a s p a ra lle l m a c h in e s , e tc .; se e th e n e w e s t p a p e r in th is a r e a [1]. A lg o r ith m s fo r F P h a v e b e e n a lso c o n s id e re d in [11, 15]. R e s e a r c h o u tc o m e s sh o w t h a t B & . B a lg o r ith m s b e c o m e u se less for m o re t h a n 10 p a r ts . S im ila rly , t h e s iz e o f M I P m o d e ls is im p r a c tic a lly la rg e e v e n fo r a s m a ll n u m b e r o f p a r ts a n d c e n te r s . T h e r e f o r e , m o r e a t t e n t i o n h a s b e e n p a id re c e n tly to t h e a p p r o x im a tio n m e th o d s . C u r r e n tly , o n ly a few c o n s t r u c t i v e a lg o r ith m s a p p lic a b le to F P a r e k n o w n , ['2, 4, 1 1, 15]. S u rp ris in g ly , u p to n o w t h e r e is n o i m p r o v i n g a lg o r ith m , a lth o u g h m a n y re c e n t p a p e r s h a v e re c o m m e n d e d th e lo c a l s e a rc h a p p r o a c h as th e m o s t p r o m is in g for v e ry h a r d o p tim iz a tio n p r o b le m s [3, 14],

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S c h e d u lin g o f fle x ib le flow lin e s a n d w o rk c e n tre s 1 2 2

2. i / | o |o s y s t e m

T h e s y s te m h a s m m a c h in e s lo c a te d in c m a c h in e c e n te r s , a n d le t A d = { 1 , . . . , m ) b e t h e s e t o f m a c h in e s , C = { l , . . . , c } b e t h e s e t o f c e n te rs . T h e c e n te r I £ C h a s m i > 1 i d e n tic a l p a r a lle l m a c h in e s , a n d le t A d; = { k / - f l , . . . , fc ,-fm /} C Ad b e t h e s e t o f m a c h in e s in t h is c e n te r , w h e re ki — £ )i= l m ,\ T h e s e t o f e p a r ts ( e le m e n ts , c u s t o m e r o r d e rs )

£ = { 1 , 2 , . . . , e } h a v e to b e p ro c e s s e d in th is s y s te m . E a c h p a r t k £ £ c o r re s p o n d s to a s e t o f Ok o p e r a t io n s O k = {f* + 1, + Ok) p ro c e s s e d in t h a t o r d e r , w h e re Ik — E ? = i °i%

a n d le t O — UJU, O k — {1, - - -, o} b e t h e s e t o f a ll o p e r a tio n s t h a t h a v e to b e p r o c e s s e d . O p e r a t i o n j c o r re s p o n d s to t h e p ro c e s s in g o f p a r t ey 1 a t c e n te r cy d u r in g a n u n i n t e r r u p t e d p r o c e s s in g t i m e py > 0 a n d c a n b e p e rfo rm e d on a n y m a c h in e fro m A d Cj. E a c h m a c h in e c a n e x e c u te a t m o s t o n e p a r t a t a tim e , eac h p a r t c a n b e p ro c e s s e d o n a t m o s t o n e m a c h in e a t a lim e , a n d e a c h p a r t k flows th r o u g h t h e s y s te m s o t h a t cy_, < cy fo r j —1 , 2 £ O k . A fe a s ib le sc h e d u le is d e fin e d by a c o u p le o f v e c to r s ( S , P ) , S — ( S i , . . . , S 0), P = ( P ] , . . . , P 0), s u c h t h a t a b o v e c o n s tr a in ts a r e s a tis fie d , w h e re o p e r a t io n j is s t a r t e d o n m a c h in e P j £ A d Cj a t tim e S j > 0.

T o p r o v id e a f o r m a l m a t h e m a ti c a l m o d e l o f th e p ro b le m w e in tr o d u c e s o m e n o tio n s . L e t £ / = { / € C? : cj = 1} b e t h e s e t o f o p e r a tio n s t h a t h a v e to b e p r o c e s s e d a t a c e n t e r I € C. B a t c h is a s u b s e t o f o p e r a tio n s p ro c e s s e d on a s e p a r a t e m a c h in e a t a c e n te r , h o w e v e r, d u e to m a c h in e id e n tity , it is n o t a s s ig n e d to a n y p a r ti c u l a r m a c h in e . T o d e te r m i n e m a c h in e w o rk lo a d , e a c h s e t C\ h a s to b e p a r ti t io n e d in to m / b a tc h e s N , C C i, i £ A d /, a n d le t n,- = |A /)|, i € M \ , I £ C. T h e batch p r o c e s s in g o r d e r is p r e s c r ib e d b y a p e r m u t a t i o n o f o p e r a t io n s it; = (tv;(1 ) , . . . , n ; ( n ,) ) £ P ( A 'i) fro m a b a tc h A ; , w h e re it,• ( A-) d e n o te s th e c le m e n t o f Afi w h ic h is in p o s itio n k in it;, a n d V ( M i ) is t h e s e t o f a ll p e r m u t a t i o n s o n t h e s e t TV;. T h e o v er a ll p ro c e ssin g o r d e r is d e fin e d b y m - tu p l e it = ( i t ] , . . . , irm ). A ll su c h p r o c e s s in g o r d e rs c r e a te t h e s e t i l = {it = ( i t , , . . . , itm) : (it, € V ( A ri), i £ A d ), ((A /;, i € A d /) is a p a r t i t i o n o f t h e s e t £ / , I £ C )}.

N e x t, l e t u s c o n s id e r r e la tio n s b e tw e e n it a n d a fe a s ib le s c h e d u le . A s s u m e t h a t it is g iv e n . F o r e a c h j £ O we d e fin e s e q u e n tia l p r e d e c e s s o r /s u c c e s s o r ¿y , s , a n d tech n o lo g ic a l p r e d e c e s s o r /s u c c e s s o r t j , i j as follow s: = it ; ( j — 1) fo r j = 2, . . . , n ; a n d & ,,(,) - o ( n u ll) ; S r i d ) = n ; ( i + 1) fo r j = l , . . . , n , - — 1 a n d s „ .(n() = o; t j = j — 1 if j > 1 a n d

'ey = k for any j € Ok-

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178 C. S m u tn ic k i

C j- i = ey, a n d fy = o o th e rw is e ; iy = j + 1 if j < o a n d eJ+J = ey, a n d iy — o o th e rw is e . F o r th is t v, s t a r t i n g tim e s Sy a n d c o m p le tio n tim e s Cy o f o p e r a tio n s h a v e to s a tis f y th e fo llo w in g c le a r c o n s tr a in ts

G , < Sy, j e O ; Lj i o , (1)

C * < Sy, j € 0 ; sy ± 0, (2)

Cy = 5 y + W , j e O . (3 )

T h e s e c o n s t r a i n t s le a d to a n o b v io u s re c u rs iv e f o rm u la e o n s t a r t i n g tim e s

Sy = max{S<^ + p Lj, S £j + pLj) (4)

w h e r e S0 = 0 = p 0, w h ic h a llo w u s to fin d t h e m in a tim e 0 ( o ) . A p p r o p r i a t e c o m p le tio n tim e s c a n b e fo u n d b y u s in g (3 ). T h e m a k e s p a n v a lu e a s s o c ia te d w ith iv w ill b e d e n o te d b y C m ax(^ )i a n d t h u s 0m&x(w) — m ax y g o tA ■

T o d e te r m i n e t h e fe a s ib le s c h e d u le fro m t h e g iv e n it 6 IT, w e h a v e to a ssig n b a tc h e s A /;, i 6 A 4 /, t o m a c h in e s i £ M i a t c e n te r /, I £ C 2. S in c e m a c h in e s a t a c e n te r a r e i d e n t i c a l, t h e a s s ig n m e n t c a n b e d o n e in a n y w ay. T h e re fo re , a fe a s ib le s c h e d u le ( S , P ) o n e c a n o b t a i n in t h e fo llo w in g m a n n e r: s e t Py = i for j £ A 'i, a n d fin d Sy b y u s in g ( 4 ) . It is c le a r t h a t a n y tv r e p re s e n ts n ? = i ( m <)' fe a sib le s c h e d u le s w ith t h e s a m e m a k e s p a n v a lu e a n d v a rio u s a s s ig n m e n ts o f b a tc h e s to m a c h in e s . F in a lly , w e c a n s t a t e o u r p r o b le m as t h a t o f f in d in g tv 6 TI w h ic h m in im iz e s C m « ( T t ) .

T h e a n a ly s is is b a s e d o n a n a u x ilia r y g r a p h m o d e l a s s o c ia te d w ith a fix e d it € II.

T h e g r a p h C7(tv) = ( 0 , A " U A ( i t ) ) h a s a s e t o f n o d e s 0 a n d a s e t o f a r c s A " U -4 (ic), w h e r e A " = U y e o - .i^ o iU y - i) } a n d A ( n ) = U y e o ;ij?! o { ( i y ,i ) } ' A rc s fro m •4* p ro c e e d fr o m c o n s tr a in ts (1 ), (3 ) a n d r e p re s e n t th e r o u te o f p a r ts th r o u g h t h e c e n te r s ; a n a rc

£ A ’ h a s w e ig h t . A rc s fro m A { v ) p ro c e e d fro m c o n s t r a i n t s ( 2 ) , (3 ) a n d r e p r e s e n t t h e p r o c e s s in g o r d e r in b a tc h e s ; e a c h su c h a r c h a s w e ig h t z e ro . E a c h n o d e j £ O r e p r e s e n ts t h e o p e r a t io n j , e v e n t " s t a r t i n g ” Sy o f th is o p e r a tio n , a n d h a s w e ig h t py. S t a r t i n g t i m e Sy e q u a l t h e le n g th o f t h e lo n g e s t p a th to n o d e j ( w ith o u t py) in 5(tv), w h e r e a s c o m p le tio n t i m e Cj e q u a l t h e le n g th o f t h e lo n g e s t p a th to n o d e j ( in c lu d in g py) in th is g r a p h . T h e m a k e s p a n C mUx(iv) e q u a ls t h e le n g th o f t h e lo n g e s t p a th ( c ritic a l p a t h ) in £/(iv).

2Batch allow us to reduce tire number of considered feasible schedules.

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S c h e d u lin g o f fle x ib le flow lin e s a n d w o rk c e n tre s l i a

L e t u s c o n s id e r a c r it i c a l p a t h in S ( t ) i n te r p r e t e d as a s e q u e n c e o f n o d e s ( o p e r a tio n s ) U = ( « i , . . . , « „ ) , Uj 6 O . W e d e fin e block B ab a s a m a x im a l s u b s e q u e n c e o f s u c c e s s iv e o p ­ e r a t io n s ( u 0, . . . , u j) fro m U su c h t h a t PU<1 = . . . = P „4. W e a lso d e n o te U = { u j , . . . , ti„ } , B ab = { n 0) ■ ■ ■ , u&), a n d for a n y j g O w e d efin e X j so t h a t itp , { x j ) = j .

T h e p r o p o s e d s o lu tio n a lg o r ith m is b a s e d on t a b u s e a rc h a p p r o a c h , w h ic h is a m o d e r n , u s e fu l te c h n iq u e fo r c o n s tr u c tin g a p p r o x im a tio n a lg o r ith m s o f v a rio u s h a r d o p tim iz a tio n p r o b le m s , [3]. T h e b a s ic id e a o f th is a p p ro a c h a p p lie d to o u r p r o b le m c o n s is ts in s t a r ti n g f r o m a n i n it i a l s o lu tio n ( f a th e r ) a n d s e a rc h in g th r o u g h i ts d e s c e n d a n ts ( t h e s e t o f s o lu tio n s c a lle d n e ig h b o r h o o d g e n e r a te d in a sp e c ific w ay fro m f a th e r ) fo r a s o lu tio n w ith t h e lo w e s t m a k e s p a n . T h e n t h e s e a rc h r e p e a ts fro m th e t h e b e s t o n e , as a n e w f a th e r s o lu tio n , a n d th e p r o c e s s is c o n tin u e d . T h e f a t h e r ’s n e ig h b o rh o o d is g e n e r a te d b y m o v es; a m o v e c h a n g e s th e p o s itio n s o f a n u m b e r o f e le m e n ts in f a th e r s o lu tio n . In o r d e r to a v o id c y c lin g , b e c o m in g t r a p p e d t o a lo ca l o p tim u m , a n d m o re g e n e ra l to g u id e t h e s e a rc h to ’’g o o d r e g io n s ” o f th e s o lu tio n s p a c e , a m e m o r y o f t h e s e a rc h h is to r y is in tr o d u c e d . A m o n g m a n y c la ss e s o f th is m e m o r y , w e re fe r o n ly to t h e ta b u lis t w h ic h re c o rd e d , fo r a c h o se n s p a n o f tim e , s e le c te d a t t r i b u t e s o f s u b s e q u e n tly v is ite d s o lu tio n s a n d / o r p e rfo rm e d m o v e s , t r e a t e d th e m a s a f o r m o f p r o h ib itio n fo r th e f u t u r e m o v es.

B a s e d on c o n c lu s io n s fro m [5, 6], w e c o n s id e r o n ly so c a lle d i n s e r t i o n m o v e s . I n o u r c a s e th is t y p e o f m o v e s is a s s o c ia te d w ith it a n d c a n b e d e fin e d b y a t r i p l e (j , i , y ) , s u c h t h a t o p e r a t io n j is re m o v e d fro m b a tc h A/p, a n d th e n in s e r te d in A/) in p o s itio n y in t h e p e r m u t a t i o n tt; (y b e c o m e s n ew p o s itio n o f j in it,). C le a rly , i t h a s to b e i E M c j , 1 V < ri,- + 1 if i jb P j , 1 < y < n ; if t' = Pj.

T h e n e ig h b o r h o o d o f it c o n s is ts o f o rd e rs it„ g e n e r a te d b y m o v e s v fro m a g iv e n s e t. B y u s i n g n a t u r a l , s im p le a p p r o a c h o n e c a n p ro p o s e t h e in s e r t i o n n e ig h b o r h o o d {tt„ : t> 6 V (ir)}

g e n e r a te d b y t h e m o v e s e t

V(*)

= u U v * ( * ) , (5 )

j£ 0 Mr,

w h e r e VJ t(ir) = (JyLV { ( j> b l/) } ' f * 7s Pi a,ltl Vy,-(it) = U y L i ^ - i ^ y j d r j- i O '. b J / ) } if * = Py T h e c o n d itio n Xj — 1 ^ y ^ x j is a d d e d to p r e v e n t r e d u n d a n c y o f m o v e s 3 . T h e s e t V j;(ir) c o n ta i n s m o v e s s u c h t h a t o p e r a tio n j is d e le te d fro m t h e p e r m u t a t i o n irp, a n d in s e r te d in a ll p o s s ib le p o s itio n s in t h e p e r m u t a t i o n -jt,-. O n e c a n v e rify t h a t t h e n u m b e r o f m o v e s in

3Moves are redundant if provide the same solutions.

(6)

1B0 C- S m n tn ic k i

V (it) e q u a ls a p p r o x im a te ly oe.

F in a l q u a li t y o f t h e ta b u s e a rc h a lg o r ith m d e p e n d s a m o n g o th e r s o n t h e c o m p u t a ti o n a l c o m p le x ity o f t h e s in g le n e ig h b o rh o o d s e a rc h a n d on t h e n e ig h b o r h o o d size. O u r p r e v io u s r e s e a r c h s h o w t h a t s u c h t h e a lg o r ith m b e h a v e s m u c h b e t t e r if w e e v a lu a te d e s c e n d a n ts d ir e c tly b y t h e c r ite r io n v a lu e , [5, 6, 7], T h e r e f o r e V (n ) s h o u ld b e g iv e n u p a s e x p e n s iv e ly tim e - c o n s u m in g , a n d w e p ro p o s e n e x t a re d u c e d n e ig h b o rh o o d .

T h e m a i n id e a o f t h e r e d u c tio n c o n s is ts in e lim in a tin g s o m e m o v e s f ro m V ( n ) fo r w h ic h i t is k n o w n a p rio ri ( w ith o u t c o m p u tin g th e m a k e s p a n ) t h a t t h e y w ill n o t i m m e d ia t e l y im p r o v e C ,na,(Tt). B y e m p lo y in g c e r ta in g r a p h p r o p e r ty ” if th e r e e x is ts in a p a th c o n ta in in g all n o d e s fro m U , w h e re U is fo u n d fo r S ( n ) , w e h a v e C nl!lx(iT„) > C max( ir ) ” w e p r o p o s e t h e fo llo w in g re d u c e d s e t o f m o v es

W ( i t ) = u U ( 6 )

icu

w h e r e W j;( ir ) C V j,(n ) a re d e fin e d o n ly for j G B at, C U in t h e fo llo w in g m a n n e r . If a = b, w e p u t W ; l (it) = 0. If i yf P j a n d a ^ b th e n TV,,• ( vr) = V ,i(n ) . In t h e r e m a in c a s e s , i.e . if i = Pj a n d a b, we s e t: kV,;(ir) = V ,,(ir) \ X j [ x Ua + — 1) if j € B„b \ {wa ,W(.}, w h e re X j ( k , t ) = \ j [ = k {[j> P j , y ) } ' , W j-,(ir) = V,-,(tv) \ X j ( l , X j ) if j = u„;

W ,,( ir ) = \ X j ( x j , m ) if j = ut,. A s tr o n g e s t r e d u c tio n is p o s s ib le for i = P j a n d a gL b: if a = 1 a n d j yt u&, w e c a n s e t W j, ( n ) = V j,(n ) \ X j ( l , x Ub — 1); b y s y m m e t r y if b = w a n d j ^ u a , w e c a n s e t = V ,,(ir) \ X j ( x u, + l , n , ) .

A ro u g h a n a ly s is o f n e ig h b o rh o o d siz e s sh o w s c le a r a d v a n ta g e s in c e |V ( n ) | / |V V (it)| ~ o / w a n d w < o. A s s u m in g u n ifo rm d i s t r ib u t i o n o f m a c h in e s o v e r c e n te r s a n d u n if o r m d i s t r ib u t i o n o f p a r t s o v e r b a tc h e s , w e h a v e o / w ~ m . T h e p ro p o s e d n e ig h b o r h o o d p o s ­ s e s s e s a ls o a c o n n e c t i v i t y p r o p e r t y w h ic h p ro v id e s fo r t a b u s e a rc h p o s s ib ility o f fin d in g o p t im a l s o lu tio n , h o w e v e r w it h o u t a s s u r a n c e t h a t it w ill b e d is c o v e re d , [7]. T h e s k ip p e d p a r t o f t h e n e ig h b o r h o o d is " le s s i n te r e s tin g ” d u e to P r o p e r t y 1.

P r o p e r t y 1. For a n y p r o c e s s in g o r d e r Tt„, v € V ( n ) \ W ( n ) , we h a v e (7max(ir„) > (7max(ir).

3 . VPI o |o a n d B \ M \ P s y s t e m s

T h e p r o d u c ti o n s y s te m h a s m a th e m a tic a l m o d e l fro m S e c tio n 2 w ith a n a d d itio n a l c o n s t r a i n t s t h a t e a c h b u ffe r Q , to t h e m a c h in e i h a s c a p a c ity ' <?, > 0, e .g . c a n s t o r e a t

(7)

S c h e d u lin g o f fle x ib le flow lin e s a n d w o rk c e n tre s i l l

m o s t <?; p a r t s a t a tim e . I t m e a n s t h a t e ac h p a r t c o m p le te d o n s o m e m a c h in e a n d d ir e c te d t o a m a c h in e i c a n b e s to r e d in th e b u ffe r <5; if it is n o t fu lfille d , o th e r w is e r e m a in s o n t h e p r im a l m a c h in e (b lo c k s th is m a c h in e ) u n til i t c a n b e t r a n s f e r r e d t o a b u f f e r . D u e to b u ffe r s e r v ic e r u le , jo b s p a s s th r o u g h t h e b u ffer Q i in t h e o r d e r g iv e n b y irt\

F i r s t , w e c o n s id e r r e la tio n s b e tw e e n a p ro c e s s in g o r d e r it 6 II a n d a fe a s ib le s c h e d u le ( 5 , P ) . L e t C j d e n o te t h e tim e o f re le a s in g t h e m a c h in e Py a f te r c o m p le tin g o p e r a t io n j . F o r a g iv e n it a p p r o p r i a te fe a sib le s c h e d u le h a v e to s a tis f y c o n s t r a i n t s ( 1 ) , (3 ) a n d a d d itio n a lly

Cj <

cj,

j e o

, (

7

)

C'lt < Sj, j

O; s j + o.

(

8

)

T o s e t b u ffe rin g c o n s tr a in ts le t u s c o n s id e r e v e n ts a s s o c ia te d w ith o p e r a tio n j — it,( x ) for s o m e 1 < x < n,-, i = Pj, su c h t h a t t j yf o. T h e r e a re tw o d is jo in c a s e s.

( a ) I f t h e b u ffe r Q j c o n ta in s a t m o s t q, — 1 (q, > 1) p a r t s a s s o c ia te d w ith r e c e n tly p e r f o r m e d o p e r a tio n s it,( x — <7, + 1) , . . . , n ,( x — 1), th e n p a r t e (j = ey c a n b e p a s s e d to t h e b u f f e r Q j i m m e d ia te ly a f te r c o m p le tio n o f o p e r a tio n t j , i.e . C'x> = C \ . I t m e a n s t h a t a ll p a r t s a s s o c ia te d w ith o p e r a tio n s ir,( 1 ) , . . . , H i(x — qi) m u s t b e ta k e n f ro m b u ffe r Q,- b e fo re tim e m o m e n t C j , th e r e f o r e , s in c e S ^ ( v) < w e h a v e

^ C,^ , X — 5i "b 1 , . . ■ , UI■ (ff)

( b ) I f t h e b u ffe r Q , c o n ta in s e x a c tly 17, (<7,- > 0) p a r ts a s s o c ia te d w ith r e c e n tly p e r fo r m e d o p e r a t io n s i r ,( x — </,•),.. . , i t ; ( x — 1), th e n p a r t ey w ill b e p a s s e d to t h e b u ffe r Q i a f te r c o m p le tio n o f o p e r a tio n tj in a tim e m o m e n t w h e n a su c c e s s iv e p a r t fro m Q , w ill b e ta k e n fo r p r o c e s s in g , i.e.

C j. = m i l l s ’, i(r_ , (x_i )} = S „ ( x - ,,) (10)

C le a rly , (1 0 ) c a n b e r e p la c e d b y m o re g e n e ra l c o n s tr a in t (9 ). T o e li m in a t e v a r ia b le s C j f r o m c o n s t r a i n t s (7) - (9), w e in tr o d u c e n o tio n o f b-se.quent.ial p r e d e c e s s o r /s u c c e s s o r ry , ry o f o p e r a t io n j d e fin e d as follow s: r „ i(j) = ir,(y —17,) for j = 17, + 1, . . . ,n ,-, a n d L*i(j) = o fo r j = 1 r,,(y ) = n » ( i + ? i) for j = 1, . . . , « ¡ -9,-, a n d 7 Il(j) = o for j = n . + g . + l , . . . , n {.

C o n s e q u e n tly , (9) c a n b e w r i t te n as S ij < . N e x t, fro m (8) w e h a v e C j < S s j if sy yt 0. T h is allo w u s to e lim in a te C j a n d to e x p re s s c o n s tr a in t (9 ) a s follow s

s Zi < s %

(11)

(8)

182. C . S m u tn ic k i

w h ic h h o ld s fo r j G O su c h t h a t r_j qb o , t j o , s ^ o. F in a lly w e o b ta in a re c u rs iv e f o r m u la e o n s t a r t i n g tim e s

w h e r e r^ = o , a0 = o, S a = 0 = pa . S im ila rly as (4 ) a ll Sy c a n b e c a lc u la te d in 0 ( 6 ) tim e . A n a p p r o p r i a te a u x ilia r y g r a p h m o d e l c an b e f o r m u la te d as fo llo w s f?(it) = ( O t A * U

• 4 (h ) U ./4.'(it)), w h e re 0 , A ' a n d A (ir) a r e d e fin e d in S e c tio n 2. T h e s e t o f a r c s - ^ ( i t ) = U j'eO ;r>5to ;ij7! o ; j , ^ o { ( l l j i \ )} r e p re s e n t b u ffe rin g c o n s tr a in ts (1 1); a n a r c ( t y , ? ^ ) 6 - 4 '( t 0 h a s w e ig h t m in u s p r .. N o tio n s 5y, Cy, a n d h a v e t h e s a m e i n te r p r e t a t i o n a s in S e c tio n 2.

B y a n a ly s in g C?(ir) o n e c a n d e te c t p o s s ib ility o f c y c le e x is te n c e . N o te , t h is h a s n o t b e p o s s ib le in s y s te m C /|o |o . C le a rly , in s u c h c a s e n o fe a s ib le s c h e d u le ( S , P ) c a n b e o b ta in e d . H e n c e , w e w ill c o n s id e r f u r t h e r o n o n ly fea s ib le overall p r o c e s s i n g o r d e r s it G n F C n , i.e.

o r d e r s w i t h o u t a c y c le in a p p r o p r i a te g r a p h Q( A).

T h u s , fo r a g iv e n re G n F , th e fe asib le s c h e d u le ( 5 , P ) c a n b e fo u n d in t h e s a m e m a n n e r a s in S e c tio n 2, h o w e v e r u s in g (1 2 ) in s te a d o f (4 ) fo r c a lc u la tin g S .

L e t u s d e fin e t h e c r itic a l p a th U a n d b lo c k s B at, in 5 ( ir ) th e s a m e w a y as in S e c tio n 2. U n f o r tu n a te ly , P r o p e r t y 1, f u n d a m e n ta l fo r th e u se o f m o v e s e t W ( r r ) , is n o t t r u e in t h i s c a s e . T h e r e f o r e , e q u a lly t h e b lo c k n o tio n a n d r e d u c e d s e t o f m o v e s a r e u s e le s s . T o r e s t o r e a ll a d v a n ta g e o u s p r o p e r tie s w e h a v e to in tr o d u c e s o m e n e w n o tio n s . T h e e x t e n d e d c r i ti c a l s e q u e n c e U " — ( u \ , . . . , i t " ) is a s e q u e n c e o b ta in e d fro m U b y in s e r tin g b e tw e e n a n y s u c c e s s iv e o p e r a tio n s u*,_i a n d u* su c h t h a t ( u f c _ i,u t) G »4* (it) a n a d d itio n a l e le m e n t s Uk.

N o te , U ' n e e d n o t b e a n y p a t h in Q (n ) . N e x t, w e d e fin e a n e x te n d e d block ZJ’ j, a s a m a x im a l s u b s e q u e n c e o f s u c c e s s iv e o p e r a tio n s (u * , . . . , « J ) fro m U ’ su c h t h a t Pu- = . . . = Pu- . F o r c o n v e n ie n c e o f n o t a t i o n s w e s e t U ‘ — { t t j , . . . , t<*}, = ( u " , . . . , u j } . F o r e a c h e x te n d e d b lo c k B *b, a n a n ti-b lo c k B b is d e fin e d as B b = () ( e m p ty ) if b = z o r ( u ?> u M-i) ^ A , a n d B b = ( j u ■ ■ ■, i , . + i ) s u c h t h a t j , = u ‘b, j k+, = sy* fo r k = 1, i = Pj, , o th e r w is e . B y t h e d e f in itio n o f >l'(iv) su c h se q u e n c e a lw a y s e x is ts a n d = f y ,. D e n o te b y B b t h e s e t o f e le m e n ts f ro m t h e s e q u e n c e B b , a n d le t U ° = U o ^ c w $ ? • W e p r o p o s e t h e fo llo w in g s e t o f m o v e s fo r ta b u s e a rc h m e t h o d

S j = m a x ^ + p t_., S , . + p , } , S r )

(

12

)

W ( * ) = U U W y , ( r r ) (13)

j e u ’U t ' .'em .

(9)

S e ts VVji(ir) C V j,(it) a r e d e fin e d d e p e n d in g on e x te n d e d b lo c k s a n d a n ti- b lo c k s , o n ly for j G B ‘b U t S b C U ' ( J U ° . If j G B ’k su c h t h a t B b = 0 w e d e fin e VVj¡(it) in e x a c tly t h e s a m e w a y a s W ji ( i r ) , se e S e c tio n 2, h o w e v e r w ith re s p e c t to e x te n d e d b lo c k B ‘b. I f j G B ^ U B g , B b ^ 0 , a n d i yt P j, w e s e t VVj,(it) = ^ ¡ ( n ) . In r e m a in c a s e s , i.e. if i = Pj, a / 6, B b yf 0 w e s e t: VV,;(ir) = if j = u j; VVj,(it) = VJ, ( i r ) \ A '; ( x u. + 1 , i uj - 1 ) if J G

r i j ii - K ) = V ji(ir) \ X j ( x ul + 1, - 1) if j G B b° \ K , r „ ; }; W ^ t c ) = V ^ r r ) \ * , ( 1 , X j ) if j = u ” ; V V j.O ) = V ;,(it) \ X j ( x j , n ; ) if j = f uj.

P r o p e r t y 2. F o r a n y p r o c e s s in g o r d e r it„ , v G V ( x ) \ V V (u), we h a v e e it h e r ir„ I I F o r C m h x { ^ v ) c i f ' m a x ( l ' ) -

T h u s , s o lu tio n m e t h o d fro m S e c tio n 2 is a p p lic a b le to th e s e s y s te m s a s w ell.

4 . C o m m e n t s a n d c o n c l u s i o n s

T h e p r o p o s e d a p p r o a c h c an b e e x te n d e d to m o re g e n e ra l cases w h ic h c o v e r m o s t o f t h e p r a c t ic a l a p p lic a tio n s in fle x ib le flow lin e s s c h e d u lin g p ro b le m s a n d a ls o in s c h e d u lin g p a r t s in a fle x ib le cell p r o d u c tio n (w o rk c e n te r s ) . O n e c a n n o te t h a t t h e fo llo w in g c o n s t r a i n t s c a n b e m o d e lle d w ith o u t e s s e n tia l c o m p lic a tio n o f t h e m o d e l: t h e r e is a j o b t r a n s p o r t t im e b e tw e e n c e n te r s , jo b s a re a rriv e d in v a rio u s tim e m o m e n ts , m a c h in e s a r e a v a ila b le a f te r i ts r e a d y t im e , jo b s m u s t b e fin ish ed b e fo re t h e g iv e n d u e d a te , jo b s m u s t b e fin is h e d j u s t in t i m e , m a c h in e s a r e n o n u n if o r m , e tc .

T h e r e m a in c o m p o n e n ts o f th e p ro p o s e d a lg o r ith m ( n o t d is c u s s e d h e r e ) a r e p r e s e n te d in d e ta il in [7] a n d h a v e b e e n s e le c te d a m o n g m a n y a lt e r n a t iv e c o n s tr u c tio n s t h a t w e re t e s te d a n d e x a m in e d in a few re c e n t p a p e rs , [5, 6, 7). E x c e lle n t c o m p u t a ti o n a l r e s u lts o b ta i n e d fo r a ll p r o b le m s fro m c ite d p a p e r s c a n b e o b ta in e d a lso in a ll c o n s id e re d c a s e s .

F u r t h e r r e s e a r c h s h o u ld b e le a d to e x a m in e t h e r e m a in ty p e s o f b u ffe rs m e n tio n e d in S e c tio n 1. P r i m a l r e s u lts o f a n a ly s is o f th e s y s te m B |C |o sh o w u se le s s o f m o d e ls b a s e d o n b a tc h p r o c e s s in g o rd e rs . In s te a d o f th e m th e m o d e ls b a s e d o n c e n t e r p r o c e s s i n g o r d e r s h o u ld b e u s e d w ith q u ite a n o th e r d e fin itio n o f m o v es.

S c h e d u lin g o f fle x ib le flow lin e s a n d w o rk c e n tre s ______________________________________

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C , S m u tn ic k i

B I B L I O G R A P H Y

1. C h e n B .: A n a ly s is o f C la sse s o f H e u ris tic s for S c h e d u lin g a T w o - S ta g e F lo w S h o p w i t h P a r a lle l M a c h in e s a t O n e S ta g e , J o u r n a l o f O p e r a tio n a l R e s e a rc h S o c ie ty , 46,

1995, 2 3 4 -2 4 4 .

2 . D in g F .Y ., K i t ti c h a r t p h a y a k D .: H e u ris tic s fo r s c h e d u lin g fle x ib le flow lin e s , C o m ­ p u t e r s I n d u s t r i a l E n g in e e r in g 26, 1994, 27-34.

3. G lo v e r F . , L a g u n a M .: T a b u S e a rc h . M o d e rn H e u r is tic T e c h n iq u e s for C o m b in a to r ia l P r o b le m s , C .R e e v e s , e d ., B la c k w e ll S c ien tific P u b lis h in g , 1993, 70-141.

4 . H u n s u c k e r J . L ., S h a h J .R .: C o m p a r a tiv e p e r fo r m a n c e a n a ly s is o f p r i o r it y r u le s in a c o n s tr a in e d flow s h o p w ith m u ltip le p ro c e s s o rs e n v ir o n m e n t. E u r o p e a n J . O p e r . R e s.

72, 1994, 102-114.

5. N o w ic k i E ., S m u tn ic k i C .: A fa s t tab u s e a rc h a lg o r ith m fo r t h e jo b s h o p . R e p o r t I C T P R E 8 /9 3 , T e c h n ic a l U n iv e rs ity of W ro c la w , 1993. M a n a g e m e n t S c ie n c e , (in p r i n t ) , 1996.

6. N o w ic k i E ., S m u tn ic k i C .: A fa s t tab u s e a rc h a lg o r ith m fo r t h e flow s h o p . R e p o r t I C T P R E 8 / 9 4 , T e c h n ic a l U n iv e rs ity o f W ro c la w , 1994. E u r o p e a n J . O p e r . R e s. (in p r i n t ) , 1996.

7. N o w ic k i E ., S m u tn ic k i C .: F low sh o p w ith p a ra lle l m a c h in e s . A t a b u s e a rc h a p p r o a c h . R e p o r t I C T P R E 3 0 /9 5 , T e c h n ic a l U n iv e r s ity o f W ro c la w , 1995.

8. S h a u k a t A . B ., H u n s u c k e r J .L .: B ra n c h a n d b o u n d a lg o r ith m fo r th e flow s h o p w ith m u lt i p le p ro c e s s o rs . E u r o p e a n J . O p e r. R e s. 51, 1991, 88-99.

9 . S r is k a n d a r a ja h C ., S e th i S .P .: S c h e d u lin g a lg o r ith m s fo r fle x ib le flo w sh o p s: W o rs t a n d a v e ra g e c a s e p e r fo r m a n c e . E u r o p e a n J . O p e r. R e s. 43, 1989, 43- 60.

10. S a n to s D .L ., H u n s u c k e r J . L ., D e al D .E .: G lo b a l lo w e r b o u n d s fo r flow sh o p s w ith m u lt i p le p ro c e s s o rs , E u r o p e a n J o u r n a l o f O p e r a tio n a l R e s e a rc h , 80, (1 9 9 5 ), 112-120.

11. S a w ik T .: A s c h e d u lin g a lg o r ith m fo r fle x ib le flow lin e s w ith lim ite d i n te r m e d i a t e d b u ffe rs . A p p lie d S to c h a s tic M o d e ls a n d D a ta A n a ly s is 9 , 1993, 127-138.

12. S a w ik T .: S c h e d u lin g fle x ib le flow lin e s w ith n o in -p ro c e s s b u ffe rs . I n t e r n a tio n a l J o u r n a l o f P r o d u c tio n R e s e a rc h , 1994, in p r i n t.

13. S m u tn ic k i C .: B lo c k a p p ro a c h in flow s h o p p ro b le m s w ith lim ite d s to r a g e s p a c e ( P o lis h ) , Z e s z y ty N a u k o w e P o lite c h n ik i Ś lą s k ie j, A u to m a ty k a 84, 1986, 2 2 3 -2 3 3 . 14. V a e s se n s R . J .M ., A a r ts E .H .L ., L c n s tr a J .K .: J o b S h o p S c h e d u lin g b y L o c a l S e a rc h ,

M e m o r a n d u m C O S O R 94-05, E id h o v e n U n iv e rs ity o f T ec h n o lo g y .

15. W it t r o c k R .J .: A n a d a p t a b l e s c h e d u lin g a lg o r ith m for fle x ib le flow lin e s . O p e r a tio n s R e s e a r c h 36, 1988, 445-453.

(11)

S c h e d u lin g o f fle x ib le flow lin e s a n d w o rk c e n tre s

R e c e n z e n t: D r h a b . in ź . E w a D y d u c h - D u d e k

W p ły n ę ł o d o R e d a k c ji do 3 0 .0 6 .1 9 9 6 r.

A b s t r a c t

T h e p a p e r d e a ls w i t h a f u n d a m e n ta l p ro b le m c o n s id e re d in fle x ib le flow l in e s c h e d u lin g . A n u m b e r o f p a r t s s h o u ld b e p e r fo r m e d in a s e q u e n c e o f p ro c e s s in g c e n te r s , w h e r e e a c h c e n t e r h a s a n u m b e r o f id e n tic a l p a ra lle l m a c h in e s , a n d t h e r e a r e i n te r m e d i a t e b u ffe rs b e tw e e n c e n te r s . A n im p r o v e m e n t a p p r o x im a tio n a lg o r ith m fo r t h is p r o b le m o f fin d in g t h e s c h e d u le w ith m in im u m n ra k e s p a n is p r e s e n te d . T h is a lg o r ith m is b a s e d o n t a b u s e a r c h a p p r o a c h w ith r e d u c e d n e ig h b o r h o o d w h ic h e m p lo y s t h e n o tio n s o f a c r itic a l p a th a n d b lo c k s o f o p e r a tio n s .

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