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ergamon

Computers ind Engng

V o l . 3 5 , N o s 3 ~ . , p p . 6 5 5 - 6 5 8 , 1 9 9 8

1 9 9 8 E l s e v i e r S c i e n c e L t d . A l l r i g h t s r e s e r v e d

P r i n t e d i n G r e a t B r i t a i n

P I I : S0360.8352 98)00182-X 0 3 6 0 - 8 3 5 2 1 9 8 1 9 . 0 0 + 0 . 0 0

R E - M A R S H A L I N G E X P O R T C O N T A I N E R S IN P O R T C O N T A IN E R T E R M I N A L S

Kap H wan Kim the corresponding au thor) and Jong Wook Bae

Depa rtment o f Industr ial Enginee ring

Pusan National University,

Research Inst i tute o f Mechanical Techno logy)

Chang jeon-dong, Kum jeong-ku, Pusan 609-735, Ko rea

FAX: 82-51-512-7603 e-mail: kapkim@hyow on.pusan, ac. kr

A B S T R A C T

In orde r to speed up the loading operat ion of export containers onto a ship, the re-marshal ing

operat ion i s an usual p ract ice in port container terminals . I t i s assumed that the current yard map for

containers is avai lable and a desi rable bay layo ut is provided. A metho dology is propo sed to convert

the curren t bay layo ut into the desi rable layou t by m oving the fewe st possible num ber o f containers

and in the shortest possible travel distance. The problem is decomposed into three sub-problems

such as the bay ma tching, the m ove planning, and the task sequencing. T he bay ma tching i s to

match a speci f ic current bay wi th a bay configurat ion in the target layout . In the move planning

stage, the numb er of containers to be mo ved from a speci f ic bay to another i s determined. The

com plet ion t ime o f the re-m arshal ing operat ion i s minimized by sequencing the m oving tasks in the

final s tage. A mathemat ical model i s suggested for each sub-problem. A numerical example i s

pro vide d to illustrate the solution pro ced ure . 1998 ElsevierScienceLtd. A ll r i g h t s reserved.

K E Y W O R D S

Re-marshaling; container port terminal; export containers; traveling salesman problem; dynamic

programming.

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

In this paper, i t is tried to r edu ce the turn-arou nd t ime o f con tainer ships in po rt con tainer terminals.

The turn-around t ime o f each ship implies the sum o f times for wai ting, berthing, unloading,

loading, and departing. On e o f important factors that effect the e ff iciency of the loading operat ion i s

the storage locat ion of export con tainers in the marshaling yard. If containers are stacked in the

sam e orde r o f loading, i t wil l spee d up the loading operation [2, 3]. Since the detai led info rma tion

about the or der of loading i s not available w hen containers s tart to arr ive at the yard , the layout o f

containers is no t ideal from the view point of the load planner wh o sequences the loading operat ion.

In this case, the re-marshal ing operat ion i s planned to convert the cu rrent layout into an ideal layout .

The re-m arshal ing operat ion i s usual ly perform ed just befo re a full am ount o f space i s al located to

the corresponding c ontainer vessel.

Suppose tha t the ideal layo ut o f con tainers for a vessel is as in Table 1. A c ontain er grou p in Table 1

implies a col lect ion o f containers of a sam e size to be loaded on to the same vessel and unload ed at

the sam e destination port . In Table 1, target bay configurations imply that a bay shou ld hav e only 23

containers o f group C and another bay should have 12 containers o f group A and 10 containers of

grou p B, and so on. N ote that since con tainers of a same group tend to be loaded successively, i t is

6 5 5

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a n u s u al p r a c t i c e t h a t c o n ta i n e r s o f m o r e t h a n t w o g r o u p s a r e n o t s t a c k e d i n a s a m e b a y t o g e t h e r .

Ho we ve r , s i nc e c on ta ine r s a r r i ve a t r a ndom po in t s o f t im e a nd the f u ll ya r d spa c e i s no t a l l oc a t e d a t

t he e a r ly s t a ge o f t he c on ta in e r a r ri va l s , t he a c tua l d i s tr i bu t ion o f c on ta ine r s m a y b e f a r f r om the

ide a l l a you t . T h e p r ob le m in t h i s pa pe r i s how to m ov e e f fi c i e n t ly c on ta ine r s t o c ha nge th e c ur r e n t

l a you t o f c on ta ine r s ( T a b le 2 ) i n to t he i de a l one ( T a b le 1 ) whic h we c a l l t he ta r ge t l a yo u t i n th i s

p a p er . T h e p r o b l e m m a y b e d e c o m p o s e d i n t o t w o f o l l o w i n g s u b - p ro b l e m s :

Table 1. An examp le of the target layout

Target bay-

configuration

number

otal

Container group

A B C D E

0 0 23 0 0

12 10 0 0 0

0 14 0 0 8

24 0 0 0 0

0 0 0 24 0

0 0 0 15 7

36 24 23 39 15

Table 2. An example of the current layout

Bay Container group

Total number A B C D E others

01 0 9 0 0 0 4

23 02 5 0 6 0 0 0

04 6 2 0 7 0 0

22

06 0 0 5 0 15 0

22 07 2 0 0 14 0 0

10 8 4 0 0 0 0

24

11 0 4 2 0 0 0

24 12 10 0 0 8 0 0

15 0 0 5 7 0 0

22

16 0 0 5 3 0 2

137 Total 31 19 23 39 15 6

Total

13

l l

15

20

16

12

6

18

12

10

133

I) Bay m atching and m ove p lann ing problem: I n o r d e r t o c o n v e r t t h e c u r r e n t l a y o u t i n t o t h e

t a r g e t l a y o u t , w e h a v e t o m a t c h e a c h c u r r e n t b a y w i t h a s p e c i f i c b a y c o n f i g u r a t i o n i n t h e t a r g e t

l a y o u t . O n c e t h e f in a l c o n f i g u r a t i o n o f e a c h b a y is d e t e rm i n e d , w e h a v e t o d e t e r m i n e h o w m a n y

c o n t a i n e r s t o b e m o v e d f r o m a b a y t h a t h a s s u r p l u s c o n t a i n e rs o f a g r o u p t o a b a y t h a t r e q u i r e s

a d d i ti o n a l c o n t a i n e r s o f t h e s a m e g r o u p .

( 2 ) S e q u e n c i n g t h e m o v i n g t a s k s : T h e m o v i n g t a sk s h a v e t o b e s e q u e n ce d i n a w a y o f m i n im i z i n g

the t o t a l t r a ve l t im e o f t r a ns f e r c r a ne s .

B A Y M A T C H I N G A N D M O V E P L A N N I N G P R O B L E M

T h e o v e r a l l s o l u ti o n p r o c e d u r e is a s f o l l o w s :

F i rs t ly , th e b a y m a t c h i n g p r o b l e m i s s o lv e d u s i n g t h e d y n a m i c p r o g r a m m i n g . B a s e d o n t h e r e s u l t o f

t h e b a y m a t c h i n g p r o b l e m , t h e m o v e p l a n n i n g i s d o n e u s i n g t h e t r a n s p o r t a t i o n p r o b l e m t e c h n i q u e .

C o n s i d e r i n g t h a t t w o t r a n s f e r c r a n e s c a n n o t p e r f o r m h a n d l i n g o p e r a t i o n s s i m u l t a n e o u s l y w i t h i n a

s p e c i f ic d i s t a n c e f r o m e a c h o t h e r , t h e t r a n s p o r t a ti o n p r o b l e m m a y r e s u lt i n a s o l u t i o n t h a t c a u s e s

i n t e r f e r e n c e b e t w e e n t r a n s f e r c r a n e s ( T C ) . T h e n , t h e c o r r e s p o n d i n g m a t c h i n g s u b s e t o f b a y s t h a t

c a u s e t h e i n t e r f e r e n c e i s a p p e n d e d t o t h e l i st o f t h e p r o h ib i te d m a t c h in g . N e x t , t h e b a y m a t c h i n g

pr ob le m i s so lve d a ga in w i th t he c o ns t r a in t tha t t he m a tc h ing in t he l is t o f t he p r o h ib i t e d m a tc h ing

s h o u l d n o t b e i n t h e s o l u t io n . T h e a b o v e p r o c e d u r e i s r e p e a te d u n t il t h e m o v e p l a n n i n g r e s u lt s i n a

f e a s ib l e so lu t ion .

U s i n g t h e d a ta in T a b l e a n d T a b l e 2 , t h e s o l u t io n o f t h e b a y m a t c h i n g p r o b l e m i s o b t a i n e d a s (0 2 ,

10 , 01 , 12 , 07 , 04) t b r t a r ge t ba y- c on f igur a t i on 1 -6 a t t he f i r st i te r a t i on a nd the nu m b e r o f t he

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S e l e c t e d p a p e r s f r o m t h e 2 2 ri d I C C I E C o n f e r e n c e

657

c o n t a i n e r s to b e m o v e d i s 6 9 . B a s e d o n t h e r e s u lt s o f t h e b a y m a t c h i n g p r o b l e m , t h e t r a n s p o r t a ti o n

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

ba ys t ha t a r e t oo c lose f r om e a c h o the r ( i n a d i s t a nc e shor t e r t ha n two ba y- l e ng th i n t h i s pa pe r ) .

W h e n w e s o l v e t h e t r a n s p o r t a t i o n p r o b l e m , a b i g v a l u e , M , i s a s s i g n e d t o t h e e n t r y i n t h e c o s t

m a t r i x c o r r e s p o n d i n g t o t h e m o v e m e n t w h i c h c a u s e s t h e i n t e r f e r e n c e b e t w e e n t w o t r a n s f e r c r an e s .

T a b l e 3 i l lu s t ra t e s t h e c o s t p a r a m e t e r s a n d t h e o p t i m a l s o l u t io n o f t h e m o v e p l a n n i n g p r o b l e m f o r

c o n t a i n e r g r o u p B f o r a g i v e n so l u t i o n o f t h e b a y m a t c h i n g p r o b l e m . T h e r e p r e s e n t s t h e d is t a n c e

f r o m b a y i t o j w h i l e x o r e p r e s e n t s t h e o p t i m a l n u m b e r o f c o n t a i n e r s t h a t s h o u l d b e m o v e d f r o m b a y

i to j . N o t e th a t x , , , i s p o s i t i v e w h i l e q ,.~ ,= M . T h u s , t h i s s o l u t io n c a n n o t b e i m p l e m e n t e d

c on s ide r in g t he phys i c a l i n t e r f e r e nc e be twe e n T Cs . I n t h is c a se , i t i s t r i e d to so lve t he ba y m a tc h ing

p r o b l e m o n e m o r e t i m e a f t e r a d d i n g t h e c u r r e n t so l u t io n o f t h e b a y m a t c h i n g p r o b l e m t o t h e l i s t o f

p r o h i b i t e d m a t c h in g . T h e n , t h e s o l u t io n s i n t h e l is t o f p ro h i b i te d m a t c h i n g w i l l b e e x c l u d e d f r o m t h e

f u r t h e r c o n s i d e r a t i o n . T h i s n e c e s s i t y t o a d d a c o n s t r a i n t i s t h e r e a s o n w h y t h e w e l l - k n o w n

H u n g a r i a n m e t h o d [ 6] i s n o t u s ed i n s o l v i n g t h e b a y m a t c h i n g p r o b l e m .

Table 3. An exam ple of infeasible movem ents or

container group B

From

01

04

10

H

Dummy

Total

To

i l

0 9 9

3 0 6

9 0 0

10 0 M

0 5 0

14 10

Total

0 9

2 2

4 4

4 4

0 5

24

Table 4. The final result of the mov e planning for

container group B

From

01

04

10

11

Dummy

Total

To

10 11

xo

9 6 10

6 0 7

0 4 M

M 0 0

0 0 0

1 1 4

Total

3 9

2 2

0 4

4 4

5 5

24

U s i n g t h e i t e r a ti v e p r o c e d u r e f o r th e b a y m a t c h i n g a n d t h e m o v e p l a n n i n g p r o b l e m , t h e f in a l

s o l u t io n o f t h e e x a m p l e i s f o u n d t o b e ( 0 2 , 1 0, 1 1 , 1 2, 07 , 0 3 ) a n d t h e n u m b e r o f c o n t a i n e r s t o b e

m o v e d i s 7 4. F o r c o n t a i n e r g r o u p B , t h e r e s u lt o f t h e m o v e p l a n n i n g i s s h o w n i n T a b l e 4 .

I n T a b l e 5 , t h e m o v i n g t a s k s f o r al l t h e c o n t a i n e r g r o u p s a r e s u m m a r i z e d w h i c h w i l l b e u s e d a s t h e

i n p u t d a t a t o t h e t a sk s e q u e n c i n g p r o b l e m .

Table 5. The list of moving asks

n, a , b , q , n , a~ b , q~ n , a , b , q , n~ a , b , q i

1 02 10 4 6 01 11 3 11 16 02 5 16 16 6 3

2 0 2 12 1 7 04 11 2 12 04 07 7 17 06 11 8

3 4 12 6 8 06 02 5 13 12 7 3

4 7 12 2 9

11 02

2 14 12 6 5

5 01 10 6 10 15 02 5 15 15 06 7

n~ : task num ber, a~ : so urce bay, b~ : d estinationbay. q, : numberof conlainers o be moved.

T A S K S E Q U E N C I N G P R O B L E M

I n th i s s ta g e , t h e m o v i n g t a s k s h a v e t o b e s e q u e n c e d i n a w a y t o m i n i m i z e t h e t o t a l tr a v e l ti m e o f

c o n t a i n e r h a n d l i n g e q u i p m e n t . N o t e t h a t s o m e m o v i n g t a s k s m a y n o t b e c a r r i e d o u t b e f o r e e n o u g h

s p a c e s b e c o m e a v a i l a b l e a t t h e d e s t i n a t i o n b a y s . T h i s p r o b l e m c a n b e d e f i n e d a s a T r a v e l i n g

S a l e s m a n P r o b l e m w i t h p r e c e d e n c e c o n s t ra i n t s w h e r e t h e p r e c e d e n c e r e la t i o n s h ip c a n n o t b e

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

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658 Selectedpapers ro m the 22nd ICC&IEConference

tO be sa t i s f i e d [ 1 , 4 , 5 ]. T h e c o nd i t i on i s tha t t he spa c e a t t he de s t i na t i on ba y shou ld b e a v a i l a b l e

b e f o r e a m o v i n g t a s k b e g i n s . T h e a v a i l ab i l it y o f t h e s p a c e f o r a t a s k d e p e n d s o n t h e p r e c e d i n g t a sk s ,

w h i c h w i l l b e p e r f o r m e d b e f o r e t h e c o r r e s p o n d i n g ta s k . T h e s o l u t i o n o f s e q u e n c i n g m o v i n g t a s k s i s

s u m m a r i z e d i n T a b l e 6 f o r t h e e x a m p l e . T h e m i n i m u m c o m p l e t i o n ti m e f o r th e r e - m a r s h a l i n g

ope ra t ion i s 107 t ime uni t s and th e opt im al task sequ enc e i s 12, 8 , 5 , 1 , 6 , 9 , 2 , 7 , 14, 17, 4 , 13, 15,

16,11, 10, 3).

Table 5. The sm mnary of sequencing mov ing asks for the example

TC TC

S eq ue n T a sk T C n u m b e r C o m p l e t i o n S e q u e n c T a sk T C n u m b e r

ce N um be nmnber a t u~ t ime e Nu m b number a t

number r at source destinafio of t a s k nu m b er er at source destinatio

n n

2 2 2 7

10 17

2

2 8 2 1 5 14 11 4 1 2

3 5 1 2 1 25 12 13 2 1

4 1 1 2 2 30 13 15 2 1

5 6 1 2 1 35 14 16 2 1

6 9 2 1 1 38 15 11 2 1

7 2 1 2 2 40 16 10 2 1

8 7 1 2 2 44 17 3 1 2

9 14 2 1 1 51

* u~j : the changeov er im e of transfer cranes to perform askj after task i

completion

u,~ time

of task

2 60

6 4

3 68

78

5 82

92

3 98

107

C O N C L U S I O N

I n th i s pa pe r , i t i s d i sc usse d how to r e - m a r sha l e xpor t c on t a ine r s i n por t c on t a ine r t e r m ina l s . T he

p r o b l e m i s d e c o m p o s e d i n t o t w o s ta g e s. I n t h e fi rs t s ta g e , t h e b a y m a t c h i n g p r o b l e m a n d t h e m o v e

p l a n n i n g p r o b l e m a r e s o l v e d s i m u l t a n e o u s ly . T h e ta s k s e q u e n c i n g p r o b l e m i s so l v e d i n t h e s e c o n d

s ta g e . B o t h t h e b a y m a t c h i n g p r o b l e m a n d t h e ta s k s e q u e n c i n g p r o b l e m a r e f o r m u l a t e d b y t h e

d y n a m i c p r o g r a m m i n g w h i l e t h e m o v e p l a n n i n g i s a p p r o a c h e d b y t h e t r a n s p o r t a t i o n p r o b l e m

t e c h n iq u e . S i n c e a ll t h e s u b - p r o b l e m s a r e s o l v e d b y m a t h e m a t i c a l p r o g r a m m i n g t e c h n i q u e s , i t t o o k

a c o n s i d e r a b l e c o m p u t a t i o n a l t i m e t o s o l v e e a c h s u b - p ro b l e m , e s p e c i a l ly f o r th e t a s k s e q u e n c i n g

pr ob l e m . He u r i s t i c t e c hn ique s ne e d t o b e de ve lo pe d f o r t he e f f i c i e n t c a l c u l a t ion .

R E F E R E N C E S

1. L . B ia nc o , A . M ingoz z i , S . R ic c i a r de l l i, a nd M . Spa d oni . E xa c t a nd he ur i s t i c p r o c e du r e s f o r t he

t r a v e li n g s a le s m a n p r o b l e m w i t h p r e c e d e n c e c o n s t ra i n t s b a s e d o n d y n a m i c p r o g r a m m i n g . IN F O R ,

32, 19-31 1994 ) .

2 . B . D . Ca s t i l ho a nd C . F . Da ga nz o . Ha ndl ing s t r a t e g i e s f o r im por t c on t a ine r s a t m a r ine t e r m ina l s .

Trans. Res.-A., 27B, 151- 166 1 993) .

3 . T . I . M ouni r a , B . D . Ca s t i l ho a nd C . F . Da ga nz o . S to r a ge spa c e vs ha nd l ing wor k i n c on t a ine r

t e r m ina l s , Trans. Re s. -A. ,27B, 13- 32 1993) .

4 . H . N . Psa r a f ti s . k - I n t e r c h a ng e p r oc e d ur e s f o r l oc a l s e a r c h i n a p r e c e de n c e - c on s t r a ine d r ou t ing

pr ob l e m . European , . Operational Res., 13 , 391- 402 1983) .

5. F. Simon. Sequ encing an d scheduling: ~4n introduction to the mathem atics o f the job-s hop , E l l i s

H o r w o o d L t d ., N e w Y o r k 1 9 8 2 ) .

6 . W. L . Winston. Operations research applications and algorithms, P W S - K E N T P u b . C o ., B o s t o n

1987) .