Heuristic Genetic Algorithms

8/3/2019 Heuristic Genetic Algorithms

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P E R G A M O N

An International Journal

computers &mathemat icsw i t h a p p l i c a t i o n s

Computers and Mathematics with Applications 44 (2002) 263-276www.elsevier.com/locate/camwa

H e u r i s t i c G e n e t i c A l g o r i t h m sfo r G e n e r a l C a p a c i t a t e d

L o t - S i z i n g P r o b l e m s

J INXING XIE* AND J IEFANG DONGDepartment of Mathematical Sciences

Tsinghua University, Beijing 100084, P.R. China

j x i e ~ m a t h . t s i n g h u a , e d u .c n

(Received January 2001; accepted February 2001)

A b s t r a c t - - T h e lot-sizing problems address the i s s u e of determining the production lot-sizes ofvarious items appearing in consecutive production stages over a given finite planning horizon. Ingeneral capacita ted lot-sizing problems, the product structure can be a general acyclic network, thecapacity constraints can be very sophisticated, and all the known parameters can be time-varying.This paper proposes heuristic genetic algorithms for these problems by designing a domain-specificencoding scheme for the lot-sizes and by providing a heuristic shifting procedure as the decodingschedule. The main contribution of these genetic algorithms is the presentation technique that encodesonly the binary variables for the setup patterns but derives other decision variables by making useof the problem-specific knowledge. Some results from the computational experiments are also given.@ 2002 Elsevier Science Ltd. All rights reserved.

Keywords--Production planning, Inventory control, Capacitated lot-sizing, Genetic algorithm(GA), Heuristics.

1 . I N T R O D U C T I O N

The widespread and popula r use of material requirements planning (MRP) systems in industryh~ resulted in increased interest in the topic of decision making in capacita ted multistage man-ufact uring systems. However, the lot-sizing procedu res used by curren tly awfilable MR P systemsare quite l imited in their abil i ty to coordinate p roduction plans of various stages of the manu-facturing processes and various capacity constraints. As firms have incorporated MRP conceptsinto their production planning and distribution systems, the capacitated multistage lot-sizingdecision problem has become a problem of prime importance.

The lot-sizing problem addresses tile issue of determining the production lot-sizes of variousitems appearing in consecutive productio n stages over a given finite planning horizon. Theobjectiv e of the problem is to reduce the total m anufacturing cost ( including setup cost, inventoryholding cost, overtiming cost, etc.) while trying to satisfy the customers ' requirements with thelimited capacity. This kind of probl em can be classified into different categorie s according to

* Author to whom all correspondence should be addressed.This research has been supported by the National Natural Science Foundation of China (Project No. 69904007).

0898-1221/02/$ - see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. Typeset by ~42~I$-TEXPII: S0898-1221(02)00146-3


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t h e p r o d u c t s t r u c t u r e s ( e .g . , s i n g le le v e l s y s t e m , s e r ia l , a s s em b l y, a n d g e n e r a l s y s t e m s ) a n d t h ec a p a c i t y s t r u c t u r e s ( e .g . , u n c a p a c i t a t e d , c a p a c i t a t e d s in g l e r e s o u rc e , a n d c a p a c i t a t e d m u l t i p l er e s o u r c e s ) . M a n y r e s e a r c h e r s h a v e s tu d i e d t h e l o t - si z in g p r o b l e m s a n d d e s i g n e d a l o t o f o p t i m a lor heur i s t i c lo t - s i z ing p roce dure s (e .g . , r e fe rences [1 9 ] a re some exce l l en t r ev iewing pa pe rs onl o t - s i z i n g p r o b l e m s ) .

Ta b l e 1 g i v e s a b r i e f r e v i e w o f s o m e i m p o r t a n t o r r e v i e w i n g l i te r a t u r e s f o r t h e d i f fe r e n t c a t e g o r i e so f t h e c a p a c i t a t e d l o t - s i z in g p r o b l e m s . B e c a u s e t h e p r o b l e m s a r e N P - h a r d a n d e v e n t h e f e a s ib i l i typ r o b l e m s w i t h s e t u p t i m e s a r e a l so N P - h a r d [ 3] , m o s t o f t h e l o t - s i zi n g p r o c e d u r e s a n d a l g o r i t h m su s e h e u r i s t i c t e c h n i q u e s t o s o l v e t h e p r o b l e m s . H o w e v e r, t h e h e u r i s t i c l o t - si z i n g t e c h n i q u e s f o rc a p a c i t a t e d p r o d u c t i o n s y s t e m s u s u a l ly c o n c e n t r a t e o n o p t i m i z i n g th e p r o d u c t i o n o p e r a t i o n ss t a g e b y s t a g e, a n d / o r o n l y c o n si d er s o m e si m p l e p r o d u c t s t r u c t u r e s a n d / o r s i m p l e c a p a c i t yc o n s t r a i n t s . F o r e x a m p l e , th e c a p a c i t a t e d l o t - s iz i n g p r o b l e m s fo r g e n e r a l p r o d u c t s t r u c t u r e s a n dm u l t i p l e r e s o u r c e s a r e s e l d o m c o n s i d e r e d . T h i s i s o b v i o u s l y a n o b s t a c l e t o t h e a p p l i c a t i o n o f t h e s el o t - s i z i n g t e c h n i q u e s i n r e a l p r o d u c t i o i i p l a n n i n g a n d s c h e d u l i n g e n v i r o n m e n t s .

Table 1. Different capacitated lot-sizing problems and so m e important/reviewingliterature.

Product Structure Uncapacitated Sin gl eResource Mul t ip leResources

Sing le Le vel [2,10,11] [12-16]

Seria l [2,17] [17]

As sem bly [2,18-20] [3,21 23]

GenerM [2,24,25] [6,261 [27,28]

I n t h e l a s t d e c a d e , t h e g e n e t i c a l g o r i t h m ( G A ) , w h i c h i s a s e a rc h t e c h n i q u e b a s e d o n t h e m e -c h a n i c s o f n a t u r a l s e l e c ti o n a n d n a t u r a l g e n e t i cs , i s r e c o g n iz e d a s a p o w e r f u l a n d w i d e l y a p p l i c a b l eo p t i m i z a t i o n m e t h o d , e s p e c i a l l y f o r g l o b a l o p t i m i z a t i o n p r o b l e m s a n d N P - h a r d p r o b l e m s [ 29 -3 1].R e c e n t l y, a l o t o f r e s e a r c h e r s s t u d i e d t h e a p p l i c a t i o n s o f G A f o r s o l v i n g t h e l o t -s i z i n g p r o b l e m sw i t h u n l i m i t e d c a p a c i t y [ 32 -3 4] a n d w i t h c a p a c i t y c o n s t r a i n t s [3 5-4 2} . N u m e r i c a l r e s u lt s o b -t a i n e d u s i n g t h e s e m e t h o d s s h o w t h a t G A ( p r o b a b l y c o m b i n e d w i t h o t h e r m e t a - h e u ri s t ic s ) i s a ne f f ec t i v e a p p r o a c h t o d e a l w i t h t h e l o t -s i z in g p r o b l e m s .

B e f o r e u s i n g G A t o s o l v e a n o p t i m i z a t i o n p r o b l e m , t h e r e a r e t w o i m p o r t a n t p o i n t s w h i c hm u s t b e a d d r e s s e d c l e ar l y : t h e f i r st is t h e e n c o d i n g ( r e p r e s e n t a t i o n ) s c h e m e f o r t h e d e c i s io nv a r i a b l e s o f t h e o p t i m i z a t i o n p r o b l e m , a n d t h e s e c o n d i s t h e e v a l u a t i o n s c h e m e f o r t h e s p e c i f ici n d i v i d u a l ( c h r o m o s o m e ) o f t h e p r o b l e m . T h e s e t w o s c h em e s a r e i n te r r e la t e d a n d t h e i r i m p r o p e rc o m b i n a t i o n c a n m a k e G A u n a b l e t o d e a l w i t h t h e o p t i m i z a t i o n p r o b l e m s e f fi c ie n tl y, e s p e c i a l l y

f o r t h e o p t i m i z a t i o n p r o b l e m s w i t h n o n t r i v i a l c o n s t r a i n t s a s i n t h e g e n e r a l c a p a c i t a t e d l o t - s i z i n gp r o b l e m s . F o r t h e c a p a c i t a t e d l o t - s iz i n g p r o b l e m s , c o n s t r a i n t s t h a t c a n n o t b e v i o l a t e d c a n b ei m p l e m e n t e d b y t h e p e n a l t y m e t h o d a n d t h e d ec o d e r m e t h o d . T h e p e n a l t y m e t h o d i m p o s e sp e n a l t i e s o n i n d i v i d u a l s t h a t v i o l a t e th e c o n s t r a i n t s , w h i l e t h e d e c o d e r m e t h o d c r e a t e s d e c o d e r so f t h e r e p r e s e n t a t i o n t h a t a v o i d c r e a t i n g i n d i v i d u a l s v i o l a t in g t h e c o n s t r a i n t s [ 3 1 ] . S i n c e e a c ho p t i m i z a t i o n p r o b l e m h a s i t s o w n f e a t u r e s , i t i s a l s o r e c o g n i z e d t h a t b e t t e r p e r f o r m a n c e c a nb e o b t a i n e d w h e n t h e p r o b l e m - s p e c i f i c k n o w l e d g e is in c o r p o r a t e d i n t o t h e s i m p l e G A [ 3 1 ]. I nf a c t , t h e r e a r e s o m e f u n d a m e n t a l l i m i t a t i o n s t o g e n e t i c a l g o r i t h m s a c c o r d i n g t o t h e s o - c a l l e dN o - F r e e - L u n c h t h e o r e m [ 43 - 45 ] . O n e o f t h e m o s t s i g n if i ca n t i m p l i c a t i o n s o f t h e N o - F r e e - L u n c ht h e o r e m i s t h a t a l g o r i t h m s s h o u ld b e m a t c h e d t o t h e s e a rc h p r o b l e m a t h a n d . " I f n o d o m a i n -s p e c i f i c k n o w l e d g e i s u s e d i n s e l e c t i n g a n a p p r o p r i a t e r e p r e s e n t a t i o n , t h e a l g o r i t h m w i l l h a v e

n o o p p o r t m f i t y t o e x c e e d t h e p e r f o r m a n c e o f a n e n u m e r a t i v e s ea r ch " [ 4 3] . G e n e r a l l y s p e a k in g ,a b e t t e r s o l u t i o n c a n b e g a in e d w i t h d e e p e r d o m a i n -k n o w l e d g e b e i n g i n c o r p o r a t e d . H o w e v e r ,t h e d o m a i n - k n o w l e d g e o f a s p e c i f i c o p t i m i z a t i o n p r o b l e m i s u s u a l l y t o o v a s t t o b e c o n s i d e r e d


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Heuristic Genetic Algorithms 265

c o m p l e t e l y w i t h a s in g l e a l g o r it h m . E v e n w o r s e ; t h e r e m a y b e s o m e d o m a i n - k n o w l e d g e w h i c h i sd i f f ic u l t t o b e c o n s i d e r e d e x p l i c i t l y w h e n d e s i g n i n g a n e v o l u t i o n a r y a l g o r i t h m . T h e s e d i f f ic u l ti e sr e v e a l t h a t w h e n d e s i g n i n g a s p e c if i c e v o l u t i o n a r y a l g o r i t h m f o r a s p e c if i c p r o b l e m d o m a i n , ac r i t ic a l i s s u e is h o w t o e x p l o i t t h e d o m a i n - s p e c i f i c k n o w l e d g e a n d h o w t o i n c o r p o r a t e i t in t o t h ec o m m o n l o g ic a n d t h e g e n e r a l s t r u c t u r e o f t h e e v o l u t i o n a r y a l g o r i t h m s .

A c c o r d i n g t o t h i s p h i l o s o p h y, a h e u r i s t i c g e n e t i c a l g o r i t h m f o r t h e g e n e r a l c a p a c i t a t e d l o t -s i z i n g p r o b l e m s i s p r e s e n t e d i n t h i s p a p e r. T h e g e n e r a l c a p a c i t a t e d l o t - s iz i n g p r o b l e m s c a n b ec h a r a c t e r i z e d b y t h e f o l l ow i n g :

( 1 ) e a c h s t a g e i n t h e p r o d u c t n e t w o r k m a y h a v e s e v e r a l p r e d e c e s s o r s a n d s e v e r a l s u c c e s s o r s ;( 2) t h e c a p a c i t y c o n s t r a i n t s m a y i n c l ud e t h e c a p a c i t y l im i t a t io n s o f n m l t i p l e r e s o u rc e s ;( 3 ) t h e c a p a c i t y o f e a c h r e s o u r c e c a n b e a d j u s t e d b y o v e r t i m i n g ;( 4 ) b e s i d e s r e s u l t i n g i n s e t u p c o s t , t h e s e t u p o f e a c h p r o c e s s c a n a l s o o c c u p y c a p a c i t y ( i. e. ,

s e t u p t i m e s a r e i n c lu d e d b u t t h e s e t u p t i m e s a r e a s s u m e d t o b e s e q u e n c e - i n d e p e n d e n t i nt h i s s t u d y ) ;

( 5) i n d e p e n d e n t d e m a n d s c a n b e g i v e n t o al l t h e i t em s i n th e p r o d u c t n e t w o r k ; a n d / o r

( 6) a ll t h e p a r a m e t e r s i n t h e p r o b l e m c a n b e ti m e - v a r y i n g .T h i s k i n d o f c a p a c i t a t e d l o t- s iz i n g p r o b l e m i s m o r e g e n e r a l t h a n t h o s e c u r r e n t l y c o n s i d e r e d b ym o s t o f t h e r e s e a r c h e rs a n d a r e m o r e a p p l ic a b l e t o re a l p r o d u c t i o n e n v i r o n m e n t s . T h e h e u r i s t i cg e n e t ic a l g o r i t h m p r o p o s e d i n t h is p a p e r a t t e m p t s t o i n c o r p o r a t e t h e p r o b l e m - s p e c if i c k n o w l e d g ei n t o t h e c o n v e n t i o n a l g e n e t ic a l g o r i t h m s a n d h e u r i st i c al l y o p t i m i z e s t h e p r o b l e m u n d e r a ll t h ec o n s t r a i n t s s i m u l t a n e o u s l y. I n o r d e r t o d o t h a t , a n e n c o d i n g ( r e p r e s e n t a t i o n ) s c h e m e fo r l o t -s i z e sa n d a s h i ft in g p r o c e d u r e w i t h p e n a l t y c o n s i d e r a t i o n s a r e d e s i g n ed b y m a k i n g u s e o f t h e c r it i ca lp r o p e r t i e s o f t h e o p t i m a l s o l u ti o n s t o t h e p r o b le m . T h i s k i n d o f m e t h o d f o r d e a l in g w i t h t h es o p h i s t i c a t e d c o n s t r a i n t s c a n b e c o n s i d e r e d a s a fu s i on o f t h e p e n a l t y m e t h o d a n d t h e d e c o d e rm e t h o d .

T h e r e m a i n d e r o f t h i s p a p e r is o r g a n i z e d a s f ol lo w s . A m a t h e m a t i c a l f o r m u l a t i o n o f t h e g e n e r a lc a p a c i t a t e d l o t - s iz i n g p r o b l e m s c o n s i d e r e d i n t h i s p a p e r i s p r o v i d e d i n t h e n e x t s e c t i o n . T h e n , i nS e c t i o n 3 , w e p r e s e n t t h e h e u r i s ti c g e n e t ic a l g o r i t h m s f o r t h e p r o b l e m s w i t h a n d w i t h o u t o v e r t i m -i n g c o n s i d e r a t i o n s , r e s p e c t i v e l y. S e c t i o n 4 g i v e s s o m e r e s u l t s o f t h e c o m p u t a t i o n a l e x p e r i m e n t s .S o m e g e n e r a l c o n c l u s i o n s a n d f u r t h e r r e s e ar c h d i r e c ti o n s a r e s u m m a r i z e d i n S e c ti o n 5 .

2 . M A T H E M A T I C A L F O R M U L A T I O N

C o n s i d e r t h e f o ll o w i ng g e n e r a l c a p a c i t a te d l o t- s iz i n g p r o b l e m ( G C L S P ) . G i v e n t h e e x t e r n a ld e m a n d f o r N i t em s o v e r a t i m e h o r i z o n o f T p e r io d s , f i n d a s o l u t io n w h i c h m i n i m i z e s t o t a ls e t u p , p r o d u c t i o n , h o l d i n g , a n d o v e r t i m i n g c o s ts , s a ti s f y i n g t h e f o l l o w i n g c o n d i t i o n s .

T h e p r o d u c t s t r u c t u r e c a n b e o r g a n iz e d a s a n a c y cl ic d i r e c te d n e t w o r k , w h e r e e v e r y n o d ei n t h e n e t w o r k i s a n i t e m a n d t h e a r c i l l u s tr a t e s t h e a s s e m b l y o r d i s t r i b u t i o n r e l a t i o nb e t w e e n i te m s , a n d t h e w e i g h t o f a n a r c is t h e q u a n t i t y r e l a ti o n b e t w e e n t h e t w o t e r m i n a ln o d e s o f t h e a r c . I n g e n e ra l p r o d u c t i o n s y s t e m s , e a c h n o d e c a n h a v e m o r e t h a n o n ei m m e d i a t e p r e d e c e s s o r a n d m o r e t h a n o n e im m e d i a t e s u c c es s o r. ( B y t o p o l o g i c a l o r d e r i n gf o r t h e d i r e c t e d g r a p h , w e c a n a s s u m e t h a t a ll t h e n o d e s a r e la b e l e d a s s a t i s f y i n g t h ec o n d i t i o n t h a t t h e l a b e l o f a p r e d e c e s s o r i s g r e a t e r t h a n t h a t o f e a c h o f i ts s u c c e s s o r s . )

T h e r e a r e m u l t ip l e r e s o u r ce s w i t h l i m i te d c a p a c i ti e s t h a t m u s t b e r e s p e c te d . W h e n n e c -e s s a r y, t h e c a p a c i t y o f a r e s o u r c e c ar l b e i n c r e a s e d b y o v e r t i m e , w h i c h i s al s o l im i t e d w i t ha m a x i m u m v a l u e .

T h e b a c k l o g g i n g o f e a c h i t e m i s n o t a l l o w e d . T h e l e a d t im e s a r e a s s u m e d t o b e c o n s t a n t a n d , w i t h o u t l os s o f g e n e r al it y, a r e a s s u m e d

t o b e z e r o .

M a t h e m a t i c a l l y, t h i s p r o b l e m c a n b e s t a t e d a s f o ll o w s .


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I i , t - 1 4 - X i t - I i t = d ,: t - k E r i j X j t ,jEs (~)

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i = 1 , . . . , N , t = 1 , . . . ,T ,

k = 1 , . . . , / ~ ' ,

i = 1 , . . , N ,

i = 1 , . . . , N ,

i = l , . . . , N ,

k = 1 , . . . , K ,

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f t h e n u m b e r o f t i m e p e r i o d s i n t h e p l a n n i n g h o r i zo n ,

I ( t h e n u m b e r o f r e s o u rc e s ,

d i t t h e e x t e r n a l d e m a n d f o r i t e m i in p e r i o d t ,

S ( i ) t h e s e t o f i m m e d i a t e s u c c e ss o r s o f i t e mi ( S ( i ) = (0 i f i i s an en d i t em ) ,

r i.j t h e n u m b e r o f u n i t s o f i t e m i r e q u i r e d t o p r o d u c e o n e u n i t o f i t e m j,

s i t t h e s e t u p c o s t f o r i t e m i i n p e r i o d t ,

c it t h e p r o d u c t i o n c o s t f o r u n i t i t e m i i n p e r i o d t ,

h i t t h e h o l d i n g c o s t f o r u n i t e n d - o f - p e r i o d i n v e n t o r y o f i t e m i i n p e r i o d t ,

o~,t t h e o v e r t i m e c o s t f o r u n i t r e s o u r c e k i n p e r i o d t ,

C k t t h e a v a i l a b l e n o r m a l c a p a c i t y o f r e s o u r ce k i n p e r i o d t ,

U~.t t h e a v a i l a b l e m a x i m u m o v e r t i m e o f r e s o u r ce k i n p e r i o d t ,

a k i t t h e c a p a c i t y n e e d e d o n r e s o u r c e k t o p r o d u c e o n e u n i t o f i t e m i i n p e r i o d t ,

At . i t t h e f i x e d lo s s o f r e s o u r c e k i n c m Te d t b r p r o d u c t i o n p r e p a r a t i o n o f i t e m i in p e r i o d t ,

a n d t h e d e c i si o n v a r i a b le s a r e

X i t t h e a m o u n t o f i t e m i p r o d u c e d i n p e r i o d t ( l ot - si z e ),

~ t a b i n a r y v a r i a b l e i n d i c a t i n g w h e r e p r o d u c t i o n is M l o w ed f o r i t e m i i n p e r i o d t ,

I i t t h e i n v e n t o r y o f it e m i a t t h e e n d o f p e r i o d t ,

O k t t h e o v e r t i m e o f re s o u r c e k i n p e r i o d t .

T h e o b j e c t i v e f u n c t i o n ( 1 ) m e a n s t o m i n i m i z e t h e t o t a l co s t w h i c h i n c l u d e s s e t u p , p r o d u c t i o n ,h o l d i n g , a n d o v e r t i m i n g c o st s . T h e c o n s t r a i n t ( 2) is t h e c o n s e r v a t i o n e q u a t i o n f o r m a t e r i a l s,a n d t h e c o n s t r a i n t ( 3 ) e n f b r c e s t h e c a p a c i t y fe a s i b il i t y. T h e c o n s t r a i n t s ( 4 ) a n d ( 6 ) i m p l y t h a tt h e s e t u p c o r r e s p o n d i n g t o a n i t e m m u s t b e p a i d b e f o re p r o d u c i n g t h e i te m . T h e n o n n e g a t i v er e s t r i c t i o n s f o r i n v e n t o r y a n d p r o d u c t i o n q u a n t i t i e s i n c o n s t r a i n t ( 5 ) a s s u re n o b a c k l o g g i n g o c c u r s,~ md t h e l a s t i n e q u a l i t y ( 7 ) s t a t e s t h e l o w e r a n d u p p e r b o u n d s a v a i l a b l e f o r o v e r t i m i n g f o r e a c hr e s o u r c e a t d i f f e r e n t t i m e p e r i o d s . M o s~ o f t h e c o n c e p t s m e n t i o n e d a b o v e c a n b e f o u n d i n m a n y

o t h e r p a p e r s on c a p a c i t a t e d l o t- s i z in g p r o b l e m s . F o r e x a m p l e , th e m a t h e m a t i c a l f o r m u l a t i o ng i v e n i n [3] i s v e r y s i m i l a r t o o u r m o d e l G C L S P e x c e p t t h a t t h d o v e r t i m e s o f r e s o u r ce s a r e n o ti n c l u d e d i n i t .


5/14

H e u r i s t i c G e n e t i c A l g o r i t h m s 2 6 7

3 . H E U R I S T I C G E N E T I C A L G O R I T H M

3 .1 . T h e E n c o d i n g S c h e m e

A g e n e t i c a l g o r i t h m o f a n o p t i m i z a t i o n p r o b l e m i s a n i t er a t iv e p r o c e d u r e a t t e m p t i n g t o h e u r is -t i c a ll y s e a r c h t h e o p t i m a l s o l u t i o n o f t h e p r o b l e m . T h r e e b a s ic g e n e ti c o p e r a t o r s i n c l u d e d i n t h ep r o c e d u r e a r e k n o w n a s r e p r o d u c t i o n , m u t a t i o n , a n d c r o s s o v er [2 9 -3 1 ]. A l l t h e s e o p e r a t o r s o p -e r a t e o n t h e c h r o m o s o m e s ( r e p r e s e n t a t i o n s o f t h e d e c i s io n v a ri a b le s ) , W h i l e d e s i g n i n g a g e n e ti ca l g o r i t h m t o s o lv e a p r o b l e m , w e m u s t f ir s t g i v e a n e n c o d i n g ( r e p r e s e n t a t i o n ) s c h e m e f o r t h ed e c i s i o n s p a c e ( s p a c e o f t h e d e c i s io n v a r i a b l es ) o f t h e p r o b l e m ,

T h e d e c i s i o n v a r i a b l e s i n G C L S P a r eX i t , ~ t , I ~ t ,a n d O k t , a m o n g w h i c h Y / t i s a 0 - 1 i n t e g e rv a r i a b le a n d t h e o t h e r s a r e p o s i t iv e v a ri a b le s o f r e al n u m b e r s . I n o r d e r t o d e s ig n a c o m p u t a t i o n a l l ye f fi c ie n t g e n e t i c a l g o r i t h m , i n t h i s s t u d y w e o n l y e n c o d e t h e s e t u p p a t t e r n s ( v a r ia b l e s. K i t) a sc h r o m o s o m e s . T h e o t h e r r e a l n u m b e r v a r ia b l esX u , I i t , a n d O at w i l l b e v i r t u a l l y c o n s i d e r e d a sb e i n g d e p e n d e n t o n Y /t, a n d t h u s , t h e y w i li b e c o m p u t e d f r o m Y /t a n d t h e k n o w n p a r a m e t e r so f t h e p r o b l em . T h e m o s t i m p o r t a n t t h i n g w h e n d e s ig n i ng s u ch a n e n c o d e > d e c o d e r s c h e m e t oG C L S P is h o w t o o b t a i n t h e v a l u e s o f t h e s e v a ri a b le s f i 'o m Y~t a n d t h e k n o w n p a r a m e t e r s . I nt h e f o l l o w i n g p a r a g r a p h s , w e w i l l s h o w h o w t h e s e v a r i a b l e s c a n b e h e u r i s t i c a l ly w o r k e d o u t b ym a k i n g u s e o f t h e p r o b l e m - s p e c i f i c k n o w l e d g e o f t h e l o t - s i z i n g p r o b l e m s .

D e n o t e t h e p o p u l a t i o n s iz e i n a g e n e t ic a l g o r i t h m a s M A X P O P ( a s s u m e t o b e a n e v e n n u m b e r )a n d t h e m a x i m u m i t e r a ti o n s ( i. e. , m a x i m u m g e n e r a t io n s ) a s M A X G E N . T h e j t h i n d i v i d u a l ( i. e. ,d e c i s i o n v a r i a b le ) i n t h e g t h g e n e r a t i o n / i t e r a t i o n i s e n c o d e d a s f o ll o w s :

yg,J = fy g , J } /~g ,J vg , J v y , J ~g , J ~g , a g ,J Y~2,g 'J " ,Y~T9 'J )~, 11 , 12 ~ ' ' '~* lT~ *21 ~ 22 , ' ' ' , 2 T , ' ' ' , Y ~ v I ~ "" - ,

j = 1 , 2 , . . . ,M A X P O P, g = 1 , 2 , . . . ,M A X G E N .( s )

A c c o r d i n g t o A f e n t a k i s a n d G a v i s h [ 2 5] , t h e r e e x i s t s t h e f o l lo w i n g p r o p e r t y ( u s u a l l y n a m e d " z e r o -s w i t c h " p r o p e r t y ) f o r u n c a p a c i t a t e d l o t -s i z i n g p r o b l e m , a s i m p l if i ed v e r s i o n o f G C L S P.

P R O P O S I T I O N . T h e r e i s a no p t i m a l s o l u ti o n t o u n c a p a c i t a t e d l o t - s iz i n g p r o b l e m ( t h e p r o b l e mG C L S P w i th o u tr e s o u r c e s c o n s t r a i n t s ) i n w h i c hX i t X i , t _ 1 ~ - O.

M a k i n g u s e o f t h is i m p o r t a n t p r o p e r t y, w e c a n h e u r is t ic a l ly c l a r i fy th e r e l a ti o n s h i p s b e t w e e n t h ed e c i s i o n v a r i a b l e s o f r e a l n u m b e r s , a n d t h e b i n a r y d e c i s i o n v a r ia b l e s}'it a n d t h e k n o w n p a r a m e t e r so f t h e p r o b l e m . G i v e n a p r o d u c t i o n p a t t e r n ( s e t u p s e q u e n c e ) , i .e ., t h e b i n a r y d e c i s io n v a r ia b l e s

{Yit, i = 1 , . . , N , t = l , . . . , T } ,

t h e p r o d u c t i o n l o t- s iz e s c a n b e d e t e r m i n e d a c c o r d i n g to t h e " z e ro - s w i tc h " p r o p e r t y a s f o ll ow s .( i) F o r a n y i t e m i a n d a n y p e r i o d r , i fgi~ = 0 , t hen Xi~ = 0 .

( i i) F or a ny i tem i and a ny p er io ds r l < r2 _< T, i f E> , = Y/-2 = 1 , an d Y/,~ = 0 for a llr l < r < 7-2, th en

X%T1 ~- E d iT ~- E " [ ' i jXjw " (9)r = ' q j 6 S ( i )

T h a t i s t o s a y, w h e n e v e r a s e t u p i s i n t r o d u c e d f o r a n i t em , w e p r o d u c e e n o u g h q u a n t i t y f o r t h i si t e m t o s a t i s f y t h e d e m a n d s o f a n i n t e g er n u m b e r o f p e r io d s , u n t il t h e p e r i o d i m m e d i a t e l y b e f o r et h e n e x t s e t u p f o r t h i s i te m . H o w e v e r, w h e n n o s e t u p i s a c t i v a t e d i n p e r i o d 1 f o r a p r o d u c t

( i te m ) , t h i s p r o c e s s i n g m e t h o d m a y l e a d t o b a c k lo g g i n g i n t h e f i rs t fe w p e r i o d s f o r t h i s p r o d u c ta n d i ts s u c c e s si v e p a r e n t p r o d u c t s . I n o r d e r t o a s s u r e t h e fe a s ib i li ty, t h e p e n a l t y m e t h o d i s u s e dt o d e a l w i t h t h e c o n s t r a i n t s t h a t n o b a c k l o g g in g is al lo w e d . T h a t is to s ay, w e c a n c h a n g e t h e


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268 J. XIE AND J. DONG

o b j e c t i v e f u n c t i o n t o i n c l u d e p e n a l t i e s fo r b a c k l o g g i n g . T h u s , t h e f i t n e s s v a l u e f o r e a c h i n d i v i d u a li s c a l c u l a t e d a c c o r d i n g t o t h e f o l l o w i n g o b j e c t i v e f u n c ti o n :

N T N T

(10 )

w h e r e ; ~ i s t h e p e n a l t y c o e f f ic i e n t ( a l a r g e e n o u g h p o s i t i v e n u m b e r ) .B a s e d o n t h e s e o b s e r v a t i o n s , th e g e n e t i c a l g o r i t h m h a s b e e n u s e d to s o l ve t h e u n c a p a c i t a t e d

l o t - s iz i n g p r o b l e m s[3>34]. F o r a c a p a c i t a t e d l o t - s iz i n g p r o b l e m , t h e " z e r o -s w i t h " p r o p e r t y n ol o n g e r h o ld s . H o w e v e r , w e c a n a c c e p t t h e s im i l a r m e t h o d o l o g y t o d e t e r m i n e t h e v a l u e s o f t h er e a l v a r ia b l e s a n d c o n c u r r e n t l y i n c l u d e o t h e r p r o c e d u r e s t o a s s u r e t h e c a p a c i t y f e a s ib i li t y. I n t h et o ll o w i n g , w e g e n e r a l i z e th e g e n e t i c a l g o r i t h m f or t h e u n c a p a c i t a t e d l o t - si z i n g p r o b l e m s t o b ec a p a b l e o f s o l v i n g G C L S P w i t h m u l t i p l e r e so u r c e s c o n s t ra i n t s a n d o v e r t i m i n g c o n s i d e r a t io n s .

3 . 2. H e u r i s t i c G e n e t i c A l g o r i t h m f o r C a p a c i t a t e d L o t - S i z i n g P r o b l e m sw i t h o u t O v e r t i m i n g

F o r c a p a c i t a t e d l o t - si z i n g p r o b l e m s , t h e o p t i m a l s o l u t io n s m a y n o t s a t i s f y t h e " z e r o -s w i t ch "p r o p e r t y. I n fa c t , t h e d e c i s i o n v a r i a b l e sX i t , Z i t a r e n o t o n l y d e p e n d e n t o n t h e p r o d u c t i o n p a t t e r n( s e t u p s s e q u e n c e) ~ t , b u t a l s o d e p e n d e n t o n t h e a w d l a b l e r e s o u rc e s c a p a c it i e sClot. I n t h i ss i t u a t i o n , w e c a n f i r s t d e t e r m i n eN i t , ~ i t f r o m Yi t w i t h o u t c o n s i d e r i n g th e r e s o u r ce s c o n s t r a i n t sa n d t a k e t h i s l o t - s i z e s p l a n a s a n i n i t i a l p l a n , w h i c h w i l l b e f u r t h e r m o d i f i e d b y c o n s i d e r i n gt h e r e s o u r c e s c o n s t r a i n t s . I n t h i s p a p e r , w e w i l l m o d i f y t h i s i n i t ia l l o t - s i z e p l a n b y " s h i f ti n g "t e c h n i q u e s t h a t h a v e b e e n w i d e l y u s e d i n t h e h e u r i s t i c s f or t h e l o t - s i z i n g p r o b l e m s [ 2 2 ,2 7 ].

T h e s h i f t i n g p r o c e d u r e c h e c k s t h e c a p a c i t y f e a s i b i l i t y o f t h e i n i t i a l l o t - s i z e s c h e d u l e f l 'o m t h el a s t p e r i o d b a c k w a r d s t o p e r i o d 1. I n e a ch p e r i o d t , a c a p a c i t y t ig h t n e s s i n d e x ,pl,.t, w h i c h i sd e f i n e d t o b e t h e r a t i o o f th e t o t a l c a p a c i t y f o r a re s o u r c e n e e d e d t o p r o d u c e a ll t h e s c h e d u l e d

i t e m s i n t h i s p e r i o d t o t h e t o t a l a w, i l a b l e c a p a c i t y f o r t h i s r e s o u r c e a t t h i s p e r i o d , i s c a l c u l a t e df o r e a c h r e s o u r c e k ,

N

C N a.,, = E ( a ~ . i t xX i t + A k ,i e x ~ ) , k = 1 , 2 , . , I ( , (11 )i = 1

C N ~ t k = 1 , 2, . , K , ( 12 )P k t - C k t ,

w h e r e C N I a t i s t h e t o t a l c a p a c i t y f o r a re s o u r c e k n e e d e d t o p r o d u c e a l l t h e s c h e d u l e d i t e m s i np e r i o d t . T h e r e m a i n i n g c a p a c i t y f o r a r es o u r c e k in p e r i o d t c a n b e c a l c u l a t e d a s

C R k t = C k t - C N k t , k = 1 , 2 , . . . , K . (13)

I t i s o b v i o u s t h a t t h e r e s o u r ce s c a p a c i t i e s a r e e n o u g h t o p r o d u c e a l l t h e s c h e d u l e d i t e m s i n ap e r i o d t i f t h e c a p a c i t y t i g h t n e s s i n d e x e s a r e a l l l e ss t h a n o r e q u a l t o o n e ( i. e .,Pt.t _< 1 for al l

k = 1 , 2 , . . . , K ) a t t h i s p e r i o d . I n t h i s c a s e , t h e s h i f t i n g p r o c e d u r e m o v e s t o p e r i o d t - 1 a n db e g i n s t o c h e c k t h e c a p a c i t y f e a s i b i l i t y i n t h i s f o l lo w i n g p e r i o d . I f i n s o m e p e r i o d t t h e t o t a lc a p a c i t y f o r a r e so u r c e n e e d e d t o p r o d u c e a l l t i l e s c h e d u l e d i t e m s i n th i s p e r i o d i s l a r g e r t h a n t h et o t a l a v a i l a b l e c a p a c i t y f o r th i s r e s o u r c e a t t h e p e r i o d ( i. e. ,Pt.t _< 1 f o r s o m e k ) , a n i n f e a s i b i l i t yo c c u r s f o r t h i s r e s o u r c e k , a n d t h u s , o n e o r m o r e o f t h e s c h e d u l e d i t e m ( s ) i n th i s p e r i o d n m s t b es h i f te d t o t i l e p r e v i o u s p e r i o d ( s ) . I f t h e r e a r e t w o o r m o r e s u c h k i n d s o f r e s o u rc e s , t h e p r o c e d u r eb e g i n s w i t h t h e r e s o u r c e w i t h t h e l a r g e st t ig h t n e s s i n d e x . I n o r d e r t o g e t a c a p a c i t y - f e a s i b l e

p l a n m o r e e f fi ci e nt ly , w e o n l y c o n s i d e r m o v i n g t h e e x c e s s iv e sc h e d u l e d p r o d u c t i o n q u a n t i t y t ot h e p e r i o d j u s t b e f o r e th i s p e r i o d . I n o r d e r t o r e d u c e a s m u c h a s p o s si b l e t h e p o s s i b i l i t y o f a n yi n f e a s i b i l i t y r e s u l t i n g f r o m t h e s h i f t i n g p r o c e d u r e t b r t h e m a t e r i a l s b e t w e e n d i f f e r e n t s t a g e s , t h e


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Heur i s t i c Gene t ic Algor i th ms 269

i t e m s a r e s h i f t e d a c c o r d i n g t o t h e d e c r e a s i n g o r d e r o f t h e l a b e l s o f t h e s c h e d u l e d i t e m s i n t h i sp e r i o d ( i. e. , fl 'o m i t e m N d o w n t o 1 ). T h i s i s b a s e d o n t h e o b s e r v a t i o n t h a t t h e i t e m w i t h a l a rg e rl a b e l w i l l n o t u s e a n i t e m w i t h a s m a l le r i n d e x a s i t s c o m p o n e n t . T h a t is t o s a y, b e g i n n i n g w i t hi t e m N a n d d o w n t o i t e m 1 , w e m o v e t h e e x c e ss i ve p r o d u c t i o n q u a n t i t y o f o n e o r m o r e s c h e d u l e di t e m s i n t h i s p e r i o d t o i t s p r e v i o u s p e r i o d u n t i l th e c a p a c i t y - u n f e a s i b i l i t y f o r t h i s r e s o u r c e i nt h i s p e r i o d i s e l i m i n a t e d . I f t h e r e i s ( ar e ) o t h e r i n f e a s ib l e r e s o u r c e ( s ) , w e r e p e a t t h i s s h i f t i n gp r o c e d u r e u n t i l a l l t h e i n f e a s i b il i ty i s e l i m i n a t e d . W h e n t h e m o v i n g p r o c e s s e s a r e f in i s h e d , t h em o d i f i e d l o t- s i z e p l a n w i l l b e a f e a s ib l e p l a n w i t h r e s p e c t t o t h e c a p a c i t y c o n s t r a i n t s . H e n c e , t h eo n l y i n f e a s i b i l i ty o f t h i s m o d i f i e d l o t- s i z e p la n m a y b e t h e b a c k l o g g i n g f o r s o m e i t e m s i n s o m ep e r i o d s , a n d t h e s e k i n d s o f i n f e as i b il it i es w i l l b e t a c k l e d w i t h t h e p e n a l t y m e t h o d b y u s i n g t h es i m i l a r p e n a l t y c o s t f u n c t i o n ( 1 0 ) f o r t h e u n e a p a c i t a t e d l o t - si z i n g p r o b l e m .

I n s u m m a r y , t h e s h i f ti n g p r o c e d u r e g o e s s u c c es s iv e ly f ro m p e r i o d T d o w n t o 1 a n d f r o m t h et i g h t e s t r e s o u r c e t o t h e m o s t f l ex i b le o n e w i t h i n a p e r i o d . W h i l e t h e c a p a c i t y i n f e a s i b il i ty o c c u r si n s o m e p e r i o d , m o v e t h e e x c e s s iv e p r o d u c t i o n o f o n e o r m o r e i t e m s i n t h i s p e r i o d j u s t b e f o r ec u r r e n t p e r i o d t o a s s u r e t h e c a p a c i t y f e a s ib i l it y ( i f t h e c u r r e n t p e r i o d i s t h e f i r st p e r i o d , t h e nt h e p r o d u c t i o n q u a n t i t y f o r t h i s i t e m i n t h i s p e r i o d w i ll b e p a r t l y o r c o m p l e t e l y l o s t ). I n t h iss h i f ti n g o r m o v i n g e x c es s iv e p r o d u c t i o n b a c k w a r d s p r o c e d u r e , t h e r e m a i n i n g c a p a c i ti e s o f a ll t h er e s o u r c e s i n t h e s e t w o s u c c e s si v e p e r i o d s a r e a d j u s t e d t h r o u g h c o n s i d e r i n g t h e s h i f te d q u a n t i t ya n d t h e p o s s i b l e e l i m i n a t e d o r a d d e d s e t u p t i m e s ( i. e. , t h e s e t u p - c o n s u m e d c a p a c i t i e s ) . T h e s e t u pp a t t e r n s o f t h e s h i f t e d i t e m s i n t h i s t w o p e r i o d s w i ll b e m o d i f i e d w h e n e v e r a p p l i c a b l e .

T h e d e t a i l e d h e u r i s t ic g e n e t i c a l g o r i t h m f o r c a p a c i t a t e d l o t- s iz i n g p r o b l em s w i t h o u t o v e r t i m i n g( G A C ) c a n b e d e s c r i b e d a s f o l l o w s .

A l g o r i t h m G A C

S T E P 1 . g : 0 . R a n d o m l y i n i ti a li z e o l d p o p ={yo, j , j = 1 , 2 , . . . , M A X P O P } , t h e p o p u l a ti o ns e t i n g t h g e n e r a t i o n .

S T E P 2 . I f g = M A X G E N , p r i n t t h e s o l u ti o n a n d s to p .

S T E P 3 . F o r ea c h y 0 ,3 E o ld p o p , j = 1 , 2 , . . . , M A X P O P, c o m p u t e i t s o b j e c t i v e fu n c t i o n v a l u eas fo l lows .

3 . 1. C o n s t r u c t t h e i n i t ia l lo t s i ze s c h e d u l e a c c o r d i n g t o e q u a t i o n ( 9 ) w i t h o u t c o n s i d e r i n g t h er e s o u r c e s c o n s t r a i n t s . C a l c u l a t eX j a c c o r d i n g t o yo, j as fo l lows.

0 j 0 , jF o r i t e m s i f l 'o m 1 t o N , lfY ~,t; = 1 , ~ t 2 = 1 (1 __ _ t l < t 2


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2 7 0 ,1 . X I E A N D J . D O N G

t h e r e m a i n i n g c a p a c i t y o f r e s o u r ce k i n p e r i o d t i s z e ro , a n d t h e p r o d u c t i o n q u a n t i t yo f i t e m i i n p e r i o d t - 1 i s i n c r e a s e d b y- C R ~ t / a a - i t . T h a t i s t o s a y, w e s e t

C R k t X j = X , j C R k t ~ O f J 1 ,X ~ t = X ~ . t 4 - - - , i , t -1 i , t-1 - - - , = 1 , CRa, t = O.

a k i t a k i t

( i i) I f C R k t + ak i tx Jt


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H e u r i s t i c G e n e t i c A l g o r i t h m s 2 71

3 . 3. H e u r i s t i c G e n e t i c A l g o r i t h m f o r C a p a c i t a t e d L o t - S iz i n g P r o b l e m sw i t h O v e r t i m i n g

For capacitated lot-sizing problems with overtiming, we can also use the shifting procedure(see [22,27]) to move the lot-sizes backwards to respect the capacity constraints similarly to

tha t of the capacitat ed lot-sizing problems without overtinfing. The only difference is tha t inthis situation, the capacity of a resource includes two parts, i.e., normal capacity and overtimecapacity. Therefore, before moving excessive pro duct ion backwards from a capacity-i nfeasibleperiod, we first consider eliminating capacity-unfeasibility by overtiming in the current period.If the sum of the normal capacity and maximum overtime capacity is still not enough to assurethe production scheduled, then apply the shifting procedure.

In order t o assure the feasibility, we change the objective functio n to including penalties asfollows:

N T K T

C O ST O (Y ,X ,I ,O ) : ~ ~ {s i t Y it + c~tX~t + h, t I i t} + ~ ~ ot: tOkt"i=1 t=l k=l e=l (14)

K T

+/3 ~ y~'[max{(), -I , t) ] ) ',k=l ~=1

where/3 is the penalty coefficient (a large enough positive number).The heuristic genetic algorithm for capacitated lot-sizing problems with overtinfing (GAO) can

be described as follows.

A l g o r i t h m G A O

Only Steps 3 and 4 of the Algorithm GAC need to be changed.

STEP 3. For each y0, j E oldpop, j = 1, 2, .. . ,MAXPOP, calculat e its objectiv e funct ion valueas follows.

3.1. Con str uct th e initial lot size schedule according to equati on (9) witho ut consi dering theresources constr aints (same as in Step 3.1 of the A lgor ithm GAC).

3.2. Elimin ate t he infeasibilities of the resources by shifting procedure similar to Step 3.2 inthe Algorithm GAC. This procedure goes from period T down to 1 and from the tightestresource to the loosest resource within a period. If the normal capac ity of a resource is notenough to eliminate the infeasibilities in a period, then first consider overtiming with thisresource in this period. If the tota l cap acity (including normal capacity and the overtime)of this resource in this period is still not enough, then shift some or all of the productionquantities to the period just before current period to assure the capacity-feasibility. (Ifthe current period is the first period, then the production for this item in this period

will be lost.) The shifting procedu re goes for item N down to 1. During the shiftingprocedure, the setup times (i.e., the setup-consumed capacities) are also considered, andthe remaining cap acit y is adjusted. The co rrespon ding setup pat ter ns y0 o, ~0.1~ 1 are alsocorrectly reset.

3.3. Deter mine in vento ry I j (the same as Step 3.3 of the A lgori thm GAC).3.4. Calculate the corresponding objective function value COSTO according to (14) from y0d ,

Xa, lJ.

STEP 4. Same as the Step 4 of the Al gori thm GAC. However, we use COS TO to evaluat e thefitness values of individuals in this algorithm instead of using COSTC in the Algorithm GAC.

4 . C O M P U T A T I O N A L E X P E R I M E N T S

The performance of the proposed algorithms GAC and GAO described in the previous sectionh ~ been evaluated on a set of testing problems. The algorithms are programmed with C + +


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2 7 2 J . X I E A N t ) J . D O N E

r u n n i n g o n a P e n t i u m - I I I . T h e e x p e r i m e n t s r e v e al th a t t h e s e a lg o r i t h m s o b t a i n a v e r y g o o d

a p p r o x i m a t i o n s o l u t i o n o f G C L S P i n r e a so n a b l e c o m p u t a t i o n t i m e . I n t h e fo l lo w i n g a r e s o m e o f

t h e e x a m p l e s o f t h e e x p e r i m e n t s .

We h a v e c o n d u c t e d a p ri r n a r y t e s t t o i n v e s t i g a t e t h e i m p a c t o f t h e c o n v ro l p a r a m e t e r s o f t h e

g e n e t i c a l g o r i th m s o n t h e p e r f o rm a n c e o f G A C a n d G A O . T h e a l g o r i th m s c o n v e r ge to a s o l u t i o n

w i t h i n 2 0 0 g e n e r a t i o n s f o r m o s t o f t h e s m a l l - s c a l e e x a m p l e s ( e .g . , N < 1 0, K < 2 ) a .n d w i t h i n 5 0 0

g e n e r a t i o n s f o r m o s t o f t h e m o d e s t e x a m p l e s ( e . g ., N ~ 2 0 , K ~ 2 ) . T h e c r o s s o v e r p r o b a b i l i t i e s

b e t w e e n 0 .6 t o 1 . 0 a n d t h e m u t a t i o n p r o b a b i l i t i e s b e t w e e n 0 .0 0 5 t o 0 .0 3 3 g i v e n o s i g n if i c a n t

d i f f er e n c e r e g a r d i n g t h e t o t a l c o s t p e r f o r m a n c e . B u t , t h e m u t a t i o n p r o b a b i l i t i e s le ss t h a n 0 .0 0 5

s h o w w o r s e t o t a l c o st p e r f o r m a n c e . I n a d d i t i o n t o th e s e c o n t r o l p a r a m e t e r s , t h e e l it i s t s t r a t e g y

c a n e n h a n c e t h e p e r f o r m a n c e s i g n if i ca n t ly , a n d t h u s , i t i s i n c o r p o r a t e d i n t o t h e a l g o r i t h m s .

I n a l l o f t h e f o l l o w i n g n u m e r i c a l e x a m p l e s , w e u s e d t h e f o l l o w i n g c o n t r o l p a r a m e t e r s f o r g e n e t i c

a l g o r i t h m s :

M a x i m u m g e n e ra t io n s M A X G E N = 50 0.

P o p u l a t i o n si ze M A X P O P = 30 .

M u t a t i o n p r o b a b i l i t y p , , = 0. 03 3 . C r o s s o v e r p r o b a b i l i t y p ~ = 0 .6 .

5 7 B o t t l e n e c k

Figure 1 . Produc t s t ruc ture fo r Example 1 .

Table 2a . Exte rn a l dem ands for end i t em and ava i lab le capac i ty fo r resource (meanvalues* ).

Perio d t 1 2 3 4 5 6

E x t e r n a l d e m a n ddl t 40 0 100 0 90 10

Av ailable capac i ty Ct 10 00 0 0 5000 5000 1000 1000

Table 2b. Setup co sts and m arginal holding costs for i tems (m ean values*) .

I tem i 1 2 3 4 5 6 7

S e t u p c o s t si t(= s i ) 400 500 1000 300 200 4 00 100

Hold ing cos thit(= hi) 12 0 .6 1.0 0.04 0.03 0.04 0.04

*Average va lues fo r t en groups of randomly genera ted da ta .

E X A M P L E 1 . C A PA C 1 TAT ED L O T- S I Z I N G W I T H O U T O V E l qT IM I N G F O R A S SE M B LY S Y S T E M . I n

t h i s e x a m p l e , t h e p r o d u c t s t r u c t u r e i s s h o w n a s F i g u r e 1 w i t h N = 7 , T = 6 . K = 1 . O n l y t h e

e n d i t e m ( i t e m 1 ) h a s e x t e r n a l d e m a n d s , a n d t h e m a r g i n a l h o l d i n g c o s t s an d s e t u p c o s t s o f a l lt h e i t e m s a r e a s s u m e d t o b e c o n s t a n t o v e r t i m e . M a r g i n a l p r o d u c t i o n c o s t s a re c o n s t a n t o v e r

t i m e a n d a r e a s s u m e d t o b e z e r o ( i. e. ,cit = 0 , V ( i , t )) . O n l y t h e p r o d u c t i o n o f i t e m s 2 a n d 3


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are capacitated, and the setup times are assumed to be ignorable ( i . e . , A ir = 0 , V ( i , ~ ) , a ~ t = 0 ,V t , Vi 2 , 3 , a 2 t = a 2 = 5 , a a t = a 3 = 8 ) . F o r t h i sexample, e a c h t e s t e d algorithm i s r u nwith ten groups of random ly generated data. The mean demands for the end item (item 1) andthe mean capacity for the resource (bottleneck) in diKe.rent time periods are shown in Table 2a.

The mean ma rginal ho lding costs and setu p costs of different items in different time perio ds areshown in Table 2b. We compared GAC with SA (simulated annealing a lgori thm [23]), TS (ta boosearch algorithm [23]), and LR (Lagrangean relaxation algorithm [21]). These algorithms areprogrammed by ourselves according to the descriptions in the original papers of Kuik e t a l . [23]and Billington e t e l . [21], respectively. For each of the rand omly gener ated problems, th e th reeprobability algorithms (i.e., algorithms GA, TS, and GAC) are run five times using differentrandom number seeds, respectively. The final results are surmnarized on Table 3. Examinationof the results shows that the solutions obtained by GAC are much better t han that of SA and TSin approximately the same computing time, while GAC obtains approximately the same bettersolutions as LR while in much less time than that, of LR.

Ta b l e 3. G A C c o m p a r e d w i t h o t h e r a l g o ri t h m s . *

A l g o r i th m M e a n O p t i m a l Va lu e C o m p a r e d w i t h G A C C o m p u t i n g T i m e ( s)

SA 10740.00 1.162 9.90

TS 9620.00 1.041 8.80

LR 9239.00 0.999 34.10

GA C 9245.00 1.000 10.10

* I n t h i s e x a m p l e , e a c h a l g o r i t h m i s t e s t e d w i t h t e n g r o u p s o f r a n d o m l y g e n e r a t e dd a t a , a n d t h e p r o b a b i l i t y a lg o r i t h m s G A , T S , a n d G A C a r e r u n fi ve t i m e s u s i n gd i f f e r e n t r a n d o m n u m b e r s e e d s .

F i g u r e 2 . P r o d u c t s t r u c t u r e f o r E x a m p l e 2 .

E X A M P L E 2 . C A PA C I TAT E D L O T- S I Z I N G W I T t t O V E RT I M I N G F O R G E N E R A L S Y S T E M . I n t h i s

e x a m p l e , t h e p r o d u c t s t r u c t u r e i s s h o w n a s F i g u r e 2 w i t h N = 2 1 , T = 6 , K = 2 . T h i s p r o b l e m

is a s i m p l if i e d v e rs i o n o f a r e al - w o r ld p r o d u c t i o n e n v i r o n m e n t c o m i n g f r o m a m a n u f a c t u r i n g

c o m p a n y . O n l y t h e e n d i t e m s 1 a n d 2 h a v e e x t e r n a l d e m a n d s , a n d t h e q u a n t i t i e s o f d e m a n d s f or

b o t h e n d i t e m s a r e n o n z e r o o n l y i n t h e l a s t p e r i o d ( t h e s e q u a n t i t i e s a r e 4 0 u n i t s a n d 1 0 u n i ts f o r

i t e m s 1 a n d 2 , r e s p e c t i v e l y, i . e . , d t 6 = 4 0 , d 26 = 1 0 ,d i t = 0 f o r o t h e r i a n d t ) . A l l t h e i t e m s h a v ei n i t i a l s t o c k s o f f iv e u n i t s ( i. e. , Ii 0 = 5 , V i ). T h e m a r g i n a l p r o d u c t i o n c o s t o f o n e u n i t o f e a c h

i t e m i s c o n s t a n t o v e r t i m e a n d i s a s s u m e d t o b e 1 0 0 ( i. e. ,c i t = 1 0 0, Vi , t ) . T h e c a p a c i t y o f b o t h


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2 7 4 J . X I E A N D J . D O N ( ;

r e s o u r c e s o c c u p i e d b y e v e r y s e t u p o f al l t h e i t e m s a r e c o n s t a n t ( i. e.,A k i t = 5 fo r a l l k , i , t ) . A l lt h e h o l d i n g c o s ts , s e tu p c o s t p a r a m e t e r s , a n d c a p a c i t y - c o n s u m i n g p a r a i n e t e r s a r e c o n s t a n t o v e rt i m e a n d a r e s h o w n i n Ta b l e 4 . T h e a v a i l a b le c a p a c i ty , m a x i m u m o v e r t i m e , a n d c o r r e s p o n d i n g

c o s t s a r e s h o w n i n Ta b l e 5 .

T a b l e 4 , S e t u p c o s t s , h o l d i n g c o s t s, a n d c a p a c i t y - c o n s u m i n g p e r u n i t .

I t e m S e t u p C o s t H o l d i n g C o s t C a p a c i t y - C o n s u m i n g C a p a c i t y - C o n s u m i n g

i 8 i t ( = 8 i ) h i t ( = h i ) a l i t ( = a l i ) a 2 i t ( = a 2 i )

1 1 2 6 0 7 6 2

2 9 7 6 5 2 1 7 6 8

3 2 1 6 0 6 5 5

4 2 5 2 0 1 4 7 6

5 2 5 2 0 1 4 3 9

6 ' 1 8 0 0 1 0 8 3

7 1 8 0 0 1 0 9 8

8 1 4 4 0 1 2 29 36 0 1 7 6

1 0 4 6 8 0 1 3 2 3

11 3 2 4 0 9 3 7

1 2 2 1 6 0 3 8 4

1 3 2 1 6 0 3 4 6

1 4 1 4 4 0 2 5 5

1 5 1 4 4 0 2 4 6

1 6 1 4 4 0 2 6 2

1 7 1 4 4 0 2 4 2

1 8 7 2 0 1 9 4

1 9 7 2 0 1 8 8

2 0 7 2 0 1 9 7

2 1 7 2 0 1 5 4

T a b l e 5 . Av a i l a b l e c a p a c i t y , o v e r t i m e , a n d c o r r e s p o n d i n g c o s t s.

P e r i o d t 1 2 3 4 5 6

N o r m a l c a p a c i t y C l t 2 0 0 0 1 5 0 0 2 0 0 0 3 0 0 0 2 0 0 0 3 0 0 0

N o r m a l c a p a c i t y C 2 t 2 0 0 0 1 5 0 0 1 8 0 0 2 4 0 0 2 0 0 0 3 0 0 0

M a z d m u m o v e r t im eU l t 1 0 0 3 0 0 2 0 0 3 0 0 1 8 0 3 5 0

M a x i m u m o v e r ti m eU2t 200 2 2 0 1 8 0 2 4 0 2 7 0 3 2 0

O v e r t i m e c o s t o l t 2 0 0 2 0 0 2 0 0 2 0 0 2 0 0 2 0 0

O v e r t i m e c o s t o2 t 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

G A C g i v e s a f e a s ib l e s o l u t i o n w i t h t o t a l c o s t o f 5 8 88 2 6 f o r th i s p r o b l e m w i t h i n t e n m i n u t e s .C h e c k i n g t h e p a r a m e t e r s o f t h e p r o b l e m , w e c a n se e t h a t t h e c a p a c i t y c o n s t r a i n t i n t h i s p r o b l e m i sv e r y t i g h t a n d G A C g i v e s a n a c c e p t a b l e s o lu t i o n . We a l s o t e s t e d t h e l o t - s i z in g p r o b l e m s w i t h t h es a m e p r o d u c t s t r u c t u r e u s in g o th e r i n p u t d a t a , a n d G A C a l w a ys o b t a in e d a c c e p t a b l e s o lu t i o n s i nr e a s o n a b l e c o m p u t i n g t i m e s . F o r t h e a l g o r i t h m s S A , T S o f [ 23 ] a n d L R o f [21 } c a n n o t d e a l w i t ht h e l o t - s i z i n g p r o b l e m s w i t h s e t u p t i m e s , w e h a v e n o t p r o v i d e d t h e s i m i l a r c o m p a r i s o n r e s u l ts a st h o s e p r o v i d e d i n E x a m p l e 1.

5 . S U M M A R Y

T h i s p a p e r d e v e l o p s a h e u r is t i c g e n e ti c a l g o r i t h m f o r g e n e r a l c a p a c i t a t e d l o t - si z i n g p r o b l e m s .T h e p r o d u c t s t r u c t u r e s a n d c a p a c i t y c o n s t r a in t s c a n b e a n y t y p e , re s o u r c es c a p a c i t y c a n b e a d -


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justed by overtiming, each item in the system can have external demands, and the cost parameters(setup costs, holding costs, production costs, overtiming costs) can be time-varying. The experi-ments show that this method is efficient and effective to solve the general capacitated lot-sizingproblems. However, the compu tationa l experiments and theoretical analyses presented in thispaper are relatively weak, and more computation experiments and theoretical analyses shouldbe made to this kind of heuristic genetic algorithm. For exanlple, optimal control parametersfor genetic algorithms such as crossover probability and nmtation probability can be identifiedby more detailed numericM experiments, and it is very attractive to study the behavior of thecost and computation time for the heuristic genetic algorithm as the complexity of the problemincreases. The heur istic al gorit hm proposed in this pap er (:an also be generalized to solve somemore complicated lot-sizing problems such as the problems with sequence-dependent setup timesand /or sequence-dependent cost parameters. Oth er genetic operators and selection strategies,such as the uniform crossover amd the ranking or tournament selection strategy, can be incor-porated into this heuristic genetic algorithm.easily. Other more complicated shifting procedurescan also be incorporated into this algorithm; for example, we can consider shutting the scheduleditems to the next earlier period that already has setups for that item. Finally, incorporating theheuristic genetic algorithm with other metaheuristics to solve the general capacitated lot-sizingand scheduling prob lems is also very attractiv e. These will be the potent ial directions for furtherresearch.

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Heuristic Genetic Algorithms