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E L S E V I E R European Journal o f Operational R esearch 84 (1995) 385-401

EUROPEAN

JOURNAL

OF OPERATIONAL

RESEARCH

A f i x e d i n t e r v a l due - da t e s c he du l i ng pr o b l e m

w i t h e a r l i ne s s a nd due - da t e c o s t s

D i l i p C h h a j e d

Depar tment of Business Adm inistration, Unil~ersity of Illinois, 350 Com merce West, Cham paign, IL 61820, USA

Received Decem ber 1992; revised August 1993

A b s t r a c t

I n th i s pa p e r we c ons ide r a p r ob le m whe r e N jobs a r e t o be s c he du led on a s ing le m a c h ine a nd a s s igne d to one o f

the two du e-da tes wh ich a re g iven a t equa l in te rva ls . Tard y jobs a re no t a llowed, thus the due-d a tes a re d eadl ines .

There i s a l inea r due-da te pena l ty and l inea r ea r l iness pena l ty , and the sum of these pena l t ie s i s to be minimized.

This problem is shown to be NP-hard . One case of the problem ( the unres t r ic ted case ) i s shown to be easy to so lve .

F o r t he r e s t r i c t e d c a se , l owe r a nd uppe r bounds a r e de ve lope d . C om pu ta t iona l e xpe r i e nc e i s r e po r t e d wh ic hsugges ts tha t the bounds a re qui te e f fec t ive .

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

T o m o t i v a t e t h e p r o b l e m i n th i s p a p e r , c o n s i d e r a s h o p w i t h a s i n g le m a c h i n e a n d N j o b s , a v a il a b l e a t

t i m e 0 , t h a t n e e d t o b e as s i g n e d d u e - d a t e s a n d s e q u e n c e d o n t h e m a c h i n e . T h e t i m e r e q u i r e d b y e a c h j o b

o n t h e m a c h i n e i s k n o w n a n d d e t e r m i n i s t i c . F i n i s h e d j o b s a r e s u p p l i e d t o t h e c u s t o m e r b y a t r u c k t h a t i s

d i s p a t c h e d a t a f ix e d t i m e i n t e r v a l, f o r e x a m p l e , a t t h e e n d o f e a c h w e e k . S i n c e a l l j o b s f i n i s h e d d u r i n g

t h e w e e k w i ll b e s h i p p e d a t th e e n d o f t h e w e e k , t h e d u e - d a t e a s s i g n e d t o a j o b w il l b e t h e e n d o f t h e

w e e k i n w h i c h t h e j o b i s s c h e d u l e d t o b e c o m p l e t e d . C u s t o m e r s p r e f e r t h e i r j o b s t o b e s h i p p e d a s ea r l y as

p o s s i b l e a n d t h u s t h e r e i s a p e n a l t y f o r a ss i g n i n g j o b s t o a d u e - d a t e t h a t i s p r o p o r t i o n a l t o t h e l e n g t h o f

t h e d u e - d a t e . A f i n i s h e d j o b i n c u r s a h o l d i n g c o s t u n ti l i t i s s h i p p e d . T h e s h o p h a s t h e p o l i c y o f d e l iv e r i n g

o n t h e p r o m i s e d d u e - d a t e , t h u s t a r d y j o b s a r e n o t a ll o w e d . T h e m e a s u r e o f c u s t o m e r s e r v ic e l e v e l i s

t h r o u g h d u e - d a t e c o st . T h e p r o b l e m i s t o f in d a n a s s i g n m e n t o f d u e - d a t e s ( t h e w e e k i n w h i ch t h e jo b w i ll

b e c o m p l e t e d ) a n d a s e q u e n c e s u c h t h a t t h e s u m o f e a r li n e s s p e n a lt y a n d d u e - d a t e p e n a l t y is m i n i m i z e d ,s u b j e c t t o n o t a r d y jo b .

T h e p r o b l e m o f s c h e d u l i n g a g i v e n se t o f j o b s t o m i n i m i z e t h e s u m o f e a r li n e s s , t a r d i n e s s , a n d

d u e - d a t e c o s t s w h e r e t h e d u e - d a t e c o s t o f a n o r d e r i s p r o p o r t i o n a l t o i ts a s s i g n e d d u e - d a t e i s w e l l

s t u d i e d . P a n w a l k a r , S m i t h a n d S e i d m a n n ( 1 9 8 2 ) c o n s i d e r t h e c a s e o f s i n g le d u e - d a t e a n d C h a n d a n d

C h h a j e d ( 19 9 2 ) c o n s i d e r t h e c a s e o f m u l t i p l e d u e - d a t e s . W e u s e t h e s a m e n o t i o n o f d u e - d a t e c o s t as

c o n s i d e r e d i n t h e s e t w o p a p e r s . M o d e l s i n w h i c h t h e r e is a si n g l e f i x ed d u e - d a t e a n d t h e o b j e c t i v e is t o

f i n d a s c h e d u l e t o m i n i m i z e t h e t o t a l e a r l i n e s s a n d t a r d i n e s s c o s t s h a v e a l s o b e e n c o n s i d e r e d i n t h e

l i t e r a t u r e ( K a n e t , 1 9 8 1 ; B a g c h i , S u l l i v a n a n d C h a n g , 1 9 8 6 ; H a l l , K u b i a k a n d S e t h i , 1 9 9 1 ) .

0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reservedS S D I

0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 2 9 1 - 5



386 D. Chhajed European Journal of Operational Research 84 (1995) 385-40l

T h e i n t e r e s t i n s t u d y i n g p r o b l e m s w i th e a r l in e s s a n d t a r d i n e s s c o s ts ( E / T C o s t ) s u p p o r t s t h e g r o w i n g

s u c ce s s a n d p o p u l a r i t y o f J u s t - i n - T i m e ( B a k e r a n d S c u d d e r , 1 9 90 ). T y p i c al l y i n a s c h e d u l e o b t a i n e d f o r a

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

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

i s s h i p p e d t o t h e c u s t o m e r , w h e r e a s a t a r d y j o b i s s h i p p e d a s s o o n a s i t i s c o m p l e t e d . I n t h e l i t e r a t u r e ,

t h e c o s t a s s o c ia t e d w i t h s h i p p in g e a c h t a r d y j o b s e p a r a t e l y is n o t c o n s i d e r e d . A c o m p r e h e n s i v e r e v i ew o f

t h e l i t e r a t u r e o n s c h e d u l i n g p r o b l e m s w i t h E / T c o s ts is g i v en in B a k e r a n d S c u d d e r ( 1 99 0 ). N o t e t h a t i n

a n J I T e n v i r o n m e n t a t a r d y j o b m a y f o r c e a c u s t o m e r t o s h u t d o w n o p e r a t i o n s , t h e c o s t o f w h i c h m a y b e

v e r y h i g h .

I n t h e p r o b l e m w e c o n s i d e r , u n l i k e t h e l i te r a t u r e c i t e d a b o v e , t h e d u e - d a t e s c a n n o t b e v i o l a t e d a n d s o

t h e y a r e d e a d l in e s a s c o n s i d e r e d i n A h m a d i a n d B a g c h i ( 1 9 86 ) a n d C h a n d a n d S c h n e e b e r g e r (1 9 88 ) . I n

t h e s e t w o p a p e r s , t h e d u e - d a t e s , w h i c h c a n b e d i s ti n c t f o r e a c h j o b a n d n o t n e c e s s a r i ly a t r e g u l a r

i n t e r v a l s , a r e k n o w n . M a t s u o ( 1 9 8 8 ) c o n s i d e r s t h e p r o b l e m w i t h f i x e d s h i p p i n g t i m e s ( s i m i l a r t o o u r s ) a n d

m i n i m i z e s t h e s u m o f o v e r t i m e a n d t a r d i n e s s c o st s . F o r e a c h j o b h e a s s u m e s t h a t a d e a d l i n e a n d a

d u e - d a t e a r e g i v e n . A j o b m u s t b e c o m p l e t e d b y i ts d e a d l i n e b u t i n c u r s t a r d i n e s s c o s t i f i t m i s s e s it s

d u e - d a t e . T h u s , f o r o u r p r o b l e m , t h e d e a d l i n e a n d t h e d u e - d a t e a r e t h e s a m e .

O u r f o c u s i n th i s p a p e r i s o n a r e s t r i c t e d v e r s io n o f t h e a b o v e p r o b l e m w h e r e o n l y tw o d u e - d a t e s a r e

c o n s i d e r e d . W e c o n c e n t r a t e o n t h i s r e s t r i c t e d b u t s t i l l u s e f u l v e r s i o n b e c a u s e e v e n t h i s c a s e i s

c o m p u t a t i o n a l l y d i f fi c u lt ( A p p e n d i x A ) . A n e x a m p l e o f t h is r e s t r i c t e d v e r s io n is a f i rm t h a t p l a n s i ts

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

m i d d l e o f t h e w e e k a n d s e c o n d a t t h e e n d o f t h e w e e k . A l s o , r e s u lt s f o r t hi s c a s e c a n b e u s e f u l i n so l v in g

t h e g e n e r a l c a s e a s w e i n d i c a t e i n S e c t i o n 5 . T h i s p a p e r a l s o p r o v i d e s a f r a m e w o r k f o r d e v e l o p i n g a

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

o u t l i n e d i n S e c t io n 5 .

T h e r e s t o f t h e p a p e r i s o r g a n i z e d a s f o ll o w s . I n S e c t i o n 2 , w e i n t r o d u c e t h e n o t a t i o n a n d f o r m u l a t e

t h e p r o b l e m f o r t h e t w o d u e - d a t e s c a s e a n d s h o w t h a t t h e r e s t r i c t e d c a s e o f t h e p r o b l e m i s e a s y t o s o lv e .

I n S e c t i o n 3 , w e p r o v i d e a m e t h o d t o o b t a i n l o w e r a n d u p p e r b o u n d s . O u r c o m p u t a t i o n a l s t u d y , r e p o r t e d

i n S e c t i o n 4 , i n d i c a t e s t h a t t h e s e b o u n d s a r e v e r y e f f e c ti v e . I n S e c t i o n 5 w e d i s c u s s p o s s i b l e u s e s o f o u r

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

A p p e n d i x A , w e s h o w t h e c o m p l e x i ty o f t h e p r o b l e m a n d p r o o f s o f se v e r a l r e s u l ts a r e i n A p p e n d i c e s B

a n d C .

2 . P r e l i m i n a r i e s

T h e s t a t e m e n t o f t h e p r o b l e m w e c o n s i d e r is: A s s i g n N j o b s t o o n e o f t w o d u e - d a t e s a n d s c h e d u l e

t h e m s u c h t h a t t h e r e is n o t a r d y jo b a n d t h e s u m o f d u e - d a t e p e n a l t i e s a n d e a r l in e s s p e n a l t i es i s

m i n i m u m . T h e t w o d u e - d a t e s a r e g i v e n a n d a r e a t e q u a l i n t e r v a l s . P r e e m p t i o n i s n o t a l l o w e d a n d o n l y

o n e j o b c a n b e p r o c e s s e d a t a ti m e . A l l c o s t f u n c t i o n s a r e a s s u m e d t o b e l in e a r . W e c a ll th i s p r o b l e m t h e2 - D u e - D a t e S c h e d u l i n g P r o b l e m ( 2 D S P ) .

W e n o w i n t r o d u c e t h e n o t a t i o n .

N o t a t i o n :

N : N u m b e r o f j ob s .

t /: P r o c e s s i n g t i m e o f j o b i .

tS: D u e - d a t e p e n a l t y p e r ti m e u n i t ( t h e e a r l i n e s s p e n a l t y p e r t i m e u n i t i s a s s u m e d t o b e o n e ) .

~-: T i m e p e r i o d ( i n t e r v a l b e t w e e n tw o s u c c e s s iv e d u e - d a t e s a n d a l so e q u a l t o t h e f i r s t d u e - d a t e ) .

Ji: J o b i . O c c a s i o n a l l y w e w i l l r e f e r t o a j o b b y i ts i n d e x , i .e . j o b i .



D. Ch hajed / European Journ al o f Operat ional Research 84 (1995) 385 -401 3 87

d x 2x

F ~ D u e - d a t e 1 ~ D u e - d a l e 2(a)

(~ x 2~

D u e - d a t e 1 ~ D u e - d a t e 2

(b)

F i g . 1 . T w o k i n d s o f s c h e d u l e s .

T : S u m o f t h e p r o c e s s i n g t i m e s o f a ll j o b s NY"i= l t i •

( a , b ) : S e t o f i n t e g e r s a , a + 1 . . . . b .

/ 2 : S e t o f a l l p o s s i b l e p e r m u t a t i o n s o f {1 . . . , N } .

Var iab les :C : C o m p l e t i o n t i m e o f j o b i.

t r = { [1 ] . . . . [ N ] } : A p e r m u t a t i o n o f { 1 , . . . , N } .

n ~: N u m b e r o f j o b s a s si g n e d t o t h e f i rs t d u e - d a t e .

I n o u r d e v e l o p m e n t , d u e - d a t e j m e a n s t h a t t h e d u e - d a t e i s j r , j = 1, 2 . W e a s s u m e t h a t T < 2 r s o

tha t t h e p r o b l e m i s f e a s ib l e . Jo bs a r e i nd e xe d su c h t ha t t~ < t 2 _< • • • < t N . G iv e n a s c h e du le , t he j ob a t

p o s i t i o n i , d e n o t e d a s Jill , r e f e r s t o t h e j o b w h i c h i s p r e c e d e d b y i - 1 j o b s . S i n c e t h e r e m a y b e i d l e t i m e ,i

t h e c o m p l e t i o n t i m e o f Jti] m a y e x c e e d E j = l t[jl .W e f ir s t n o t e t h e f o l l o w i n g o b s e r v a t io n s . T h e r e m a y b e i d l e t im e b e f o r e t h e s t a r t t i m e o f t h e f i rs t j o b s

a s s i g n e d t o d u e - d a t e s 1 a n d 2 . H o w e v e r , t h e r e w i ll b e n o i d l e ti m e b e t w e e n j o b s a s s i g n e d t o t h e s a m e

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

s h i ft i n g th e j o b s t o t h e r ig h t . A l s o t h e c o m p l e t i o n t i m e o f t h e l a s t j o b a s s i g n e d t o t h e f ir s t d u e - d a t e m a y

n o t c o i n c i d e w i t h th e d u e - d a t e . T h e d u e - d a t e o f J [ i ] wi l l n o t b e m o r e t h a n t h e d u e - d a t e o f J [i+ q . T h u s , i n

o r d e r t o f i n d a n o p t i m a l s o l u t i o n w e o n ly n e e d a p e r m u t a t i o n o f j o b s a n d n l , t h e n u m b e r o f j o b s a s s ig n e d

t o t h e f i rs t d u e - d a t e . A l l jo b s { Jtll . . . . . J [n ,j } w i ll h a v e d u e - d a t e r w h i l e t h e r e m a i n i n g j o b s w i ll h a v e

d u e - d a t e 2 r . I t is o p t i m a l t o s c h e d u l e j o b s a s s ig n e d t o t h e s a m e d u e - d a t e i n n o n - i n c r e a s i n g o r d e r o f th e i r

p r o c e s s i n g t im e . F ig . 1 s h o w s t h e n a t u r e o f th e t w o k i n d s o f s c h e d u l e s t h a t m a y o c c u r. W e w i ll d e n o t e t h e

o p t i m a l v a l u e o f a fu n c t i o n Z b y Z * .

W i t h t h e s e n o t a t i o n s , 2 D S P c a n b e f o r m u l a t e d a s :

( P I )

n I

m i n Z = 2 ~ ( r - C[i]) +

o ' G O i = 1

s . t . C [ i ] >- C [ i - u + t [ i ] ,

C[nl ] <- 7" ,

C [ N ] ~ 2 r ,

n l ~ ( 1 , N ) .

N

Ei =n I + 1

( 2 r - C t i l ) + ( n , + 2 ( N - n , ) ) r 6

i = 1 . . . . . N , (1 )

(z )

(3 )

( 4 )

H e r e C [0 = 0 . T h e f i r s t t e r m i n t h e o b j e c t i v e f u n c t i o n s u m s u p t h e e a r l i n e s s p e n a l t y o f t h e f i r s t n~ j o b s

t h a t a r e a s s i g n ed t h e d u e - d a t e r . T h e s e c o n d t e r m is t h e e a r l i n es s p e n a l t y o f th e jo b s a s s i g n e d t h e



388 D. C hhajed / E uropea n Journal o f O pera t ional Research 84 (1995) 385-40 1

~=0 t=5.5 t '= l 1

F i g . 2 . A C a s e I I op t i m a l s o l u t i on .

d u e - d a t e 2 r a n d t h e l a s t t e r m is th e d u e - d a t e p e n a l t y . A n y o p t i m a l s o l u t i o n t o 2 D S P w i ll sa t is f y o n e o f

t he fo l l owi n g t wo c a se s (F i g . 1 ) :

C a s e I : Ctn~] = r .

C as e II: CI~ d < ~-.

F i g . 2 show s a n e xa m pl e o f Ca se I I op t i m a l so l u t i on . In t h i s e xa m pl e , N = 6 , ~" = 5 .5 , t~ = { 1, 1 , 1 , 1 , 3 , 3}

a nd 3 = 1 .

W e n o w p r e s e n t a s u f f ic i e n t c o n d i t i o n w h e n t h e r e e x is ts a C a s e I o p t i m a l s o l u ti o n .

L e m m a 1. I f 2~ " - T > t N , t h e n t h e r e e x i s ts a n o p t i m a l s o l u t i o n t h a t i s C a s e I .

P r o o f . S u p p o s e t h e c o n d i t i o n o f t h e L e m m a i s s a ti s fi e d a n d w e h a v e a C a s e I I o p t i m a l s o l u t i o n w i th j o bJ k a s t h e f ir s t j o b a s s i g n e d t o t h e s e c o n d d u e - d a t e . T h u s J~ s t a r t s b e f o r e t h e f ir s t d u e - d a t e b u t g e t s

c o m p l e t e d a f t e r r ( F i g . 3 a ) . S i n c e t k < t N , w e c a n a s s i g n Jk t o d u e - d a t e 1 , m o v e J k t o t h e l e f t m o s t

p o s i t i o n s o t h a t i t is th e f i r s t j o b p r o c e s s e d a n d s t il l g e t a f e a s i b l e s o l u ti o n . W e n o w m o v e a l l t h e j o b s

a s s i g n e d t o t h e f i r s t d u e - d a t e t o t h e r i g h t u n t i l t h e c o m p l e t i o n t i m e o f t h e l a st j o b a s s i g n e d t o t h e f i r s t

d u e - d a t e i s r ( F i g . 3 b ) . U s i n g t h e n o t a t i o n d e f i n e d i n F ig . 3, t h e c h a n g e i n t h e c o s t a s a r e s u l t o f t h i s s h if t

is

Z I = T 1 - T 2 + tk - r n l - r ~ ,

w h e r e n 1 is t h e n u m b e r o f j o b s c o m p l e t i n g b e f o r e r i n th e C a s e I I s o l u t io n . O r ,

A = T - 2 T 2 + t k - r n 1 - ~ 3

< 2 r - 2 T 2 - r n l - r 6 ( s i n c e T + t k < _ 2 r )

< - r ( 2 + n l ) -~ - ~ .

T h u s t h e n e w C a s e I s o l u t i o n r e s u l t s i n a n e t d e c r e a s e i n t h e c o s t. [ ]

W e n o w a n a l y z e t h e t w o c a s e s s e p a r a t e l y .

C a s e I : Ctn 0 = ~ '. In t h i s c a se a n op t i m a l so l u t i o n wi l l s a t is fy t he fo l l ow i ng p rop e r t y :

d

1 T 1 - -~ , r ~ - - T 2 :,

t ~

~ 2x

(a)

2~( b )

F i g . 3 . F i g u r e f o r L e m m a 1 .



D . C h h a j e d / E u r o p e a n J o u r n a l o f O p e r a ti o n a l R e s e a r c h 8 4 (1 9 9 5 ) 3 8 5 - 4 0 1 3 8 9

P r o p e r t y 1 . I f a n o p t i m a l s o l u t i o n t o (Px) sat is f ies C[nl] = ~', t he n id l e t ime c an be on ly be fore the f i r s t j ob

t h a t i s p r o c e s s e d i n d u e - d a t e 1 ( l e f l m o s t j o b i n F i g. 1 a n d t h e f i r s t j o b ( p o s i t i o n n I + 1) i n d u e - d a t e 2 .

F o r t h i s c a s e C[n,] = ~" a n d C[N 1 = 2 z a n d t h e o b j e c t i v e f u n c t i o n o f ( P~ ) c a n b e r e w r i t t e n a s

n I N

E ( C [ n , l - C t i ] ) + E ( C i N l - C t i l ) + ( Z U - n l ) ~'6i = 1 i = n l + l

n l n l N N

= ~ ~ t u l + Y'~ ~ t u l + ( Z N - n l ) ~ ' t 5i = 1 j = i + l i = n l + l j = i + l

n~ N

= Y ' ~ ( i - 1 ) t t i l + E ( i - l - n l ) t t i l + ( 2 N - n , ) r 6i = 1 i = n l + l

n~ N - n t

= E ( i - 1) t l i I + Y '. ( i - 1) t [ i+n,] + (2 N - n l ) r 3 . (5 )i = 1 i = 1

T h e r e f o r e ( P 0 , w i t h t h e a d d i t i o n a l c o n s t r a i n t t h a t C [ , ,] = ~', c a n b e f o r m u l a t e d a s ,

( P 2 )

n I N - n j

m i n Z l = ~ _ , ( i - 1 ) t t i ] +o '~ = Q i = 1 i = l

s. t. C[1 ] > t[]],

C [ i] = C [ i - 1] --t- t [ i ] ,

( i - 1 ) t[ i+ . ,l + ( 2 N - n I ) T 6

i = 2 . . . . . n ~, n I + 2 , n~ + 3 . . . . , N ,

C [ n l + i ] > _ '7 - q - t [ n l + 1 ] ,

CIn,I = r , an d (4 ) .

T h e n u m b e r m u l t i p l i e d t o t h e p r o c e s s i n g t i m e o f a j o b a s s i g n e d t o p o s i t io n i is t h e p o s i t i o n a l w e i g h t o f

p o s i t i o n i . I n ( P2 ), f o r e x a m p l e , t h e p o s i t i o n a l w e i g h t o f p o s i t i o n i ( i < n ~ ) i s i - 1 a n d t h e p o s i t i o n a lwe igh t o f po s i t ion i ( i > n l ) i s i - 1 - n~

C a s e H : C[ , ,I < ~-. A l l op t im al so lu t ion s un de r th i s case s a t i s fy the fo l lowin g p roper ty :

P r o p e r t y 2 . T he e n t i r e id l e t ime in th i s c ase i s be fore the job in pos i t i on 1 . T he s ta r t t ime fo r the f i r s t j ob i s

2 ~ ' - T .

T h e o b j e c t i v e f u n c t i o n o f ( P 1 ) c a n b e s i m p l i f i e d a s

N

] ~ ( i - 1 )t[i] + ( 2 N - n l ) ~ ' 6 - n l~ '. ( 6 )i = 1

N o t e t h a t i n t hi s c a se t h e s u m o f t h e p r o c e s s in g t i m e s o f th e j o b s a s s ig n e d t o t h e s e c o n d d u e - d a t e w i ll

e x c e e d z . T h e f o r m u l a t i o n ( P~ ) n o w t a k e s t h e f o l l o w i n g f o r m .

( P 3 )

N

m i n Z 2 = Y'~ ( i - 1 ) t[ il + ( 2 N - n l ) ~ ' 6 -- n l~ -o ' ~ Q i = 1

s.t . C[1 ] = tflI + 2~- - T ,

C [ i ] = C [ i _ l ] q - t [ i ] , 1 = 2 , . . . , N ,

C[n,] < z , a n d ( 4 ) .



3 9 0 D . C h h a j e d / E u r o p e a n J o u r n a l o f O p e r a t i o n a l R e s e a r c h 8 4 ( 1 9 9 5 ) 3 8 5 - 4 0 1

I n o r d e r t o s o l v e ( 2 D S P ) , i f L e m m a 1 i s s a t is f i e d , w e s o l v e ( P 2 ) a n d s e t Z * = Z * . O t h e r w i s e , w e s o l ve

( P2) a nd ( P3) a nd se t

Z * = m i n { Z ~ ' , Z ~ ' } . ( 7 )

I n A p p e n d i x A w e s h o w t h a t ( P 1 ) i s b i n a r y N P - h a r d . H o w e v e r , t h e r e i s a c la s s o f p r o b l e m s w h i c h i s

e a s y t o s o lv e . T h i s c l a s s c o r r e s p o n d s t o t h e c a s e w h e n T < r , a n d i s c a l l e d t h e u n r e s t r i c t e d c a s e . T h e

u n r e s t r i c t e d c a s e i s s i m i l a r t o t h e c a s e c o n s i d e r e d b y K a n e t ( 1 9 8 1 ) f o r t h e m e a n a b s o l u t e d e v i a t i o n

p r o b l e m . S i n c e th e s u m o f th e p r o c e s s in g t i m e s o f j o b s a s s i g n e d t o t h e s e c o n d d u e - d a t e c a n n o t e x c e e d ~',

a n o p t i m a l s o l u t io n t o t h e u n r e s t r i c t e d c a s e w il l c o r r e s p o n d t o C a s e I . A n y n u m b e r o f j o b s c a n b e

a s s ig n e d t o t h e f ir s t d u e - d a t e w i t h o u t t h e s u m o f t h e i r p r o c e ss i n g t i m e s e x c e e d i n g t h e l e n g t h o f th e

p e r i o d . I t i s t h i s o b s e r v a t i o n w h i c h m a k e s t h i s p r o b l e m e a s y t o s o lv e . T h u s , s t a r t i n g w i t h n ~ = m 0 t o N ,1

w h e r e m 0 = m i n i m u m i n t e g e r > ~ N , w e f i rs t fi n d t h e p o s i t i o n a l w e i g h t s . A f t e r s o r t i n g t h e p o s i t i o n a l

w e i g h t s i n d e c r e a s i n g o r d e r t h e o p t i m a l a s s i g n m e n t o f j o b s , w i t h f i x e d n ~ , i s t o a s s i g n j o b i t o p o s i t i o n k

w h e r e t h e p o s i t i o n a l w e i g h t o f p o s i t i o n k h a s r a n k i . S o l u t i o n t o 2 D S P is t h e s o l u t i o n f o r t h a t v a l u e o f n 1

t h a t g i v e s t h e m i n i m u m c o s t .

I n t h e n e x t s e c t io n , w e c o n s i d e r t h e c a s e w h e n T > r .

3 . L o w e r a n d u p p e r b o u n d s

I n t h is s e c t io n w e p r o v i d e a m e t h o d t o o b t a i n l o w e r a n d u p p e r b o u n d s f o r t h e o b j e c t iv e f u n c t i o n o f

( P 1 ) w h e n 2 z >_ T > ~'. C o n s i d e r a c o n s t r a i n e d v e r s i o n o f t h e p r o b l e m w h e n n 1 i s p r e s p e c i f i e d . L e t t h e

o p t i m a l o b j e c ti v e f u n c t i o n v a l u e o f t h i s c o n s t r a i n e d p r o b l e m ( w i th n 1 j o b s a s s ig n e d t o d u e - d a t e 1 ) b e

Z * ( n ~ ) . A s t h e n u m b e r o f j o b s a s s ig n e d t o t h e f ir s t d u e - d a t e c a n v a r y f r o m 1 t o N ( m o r e o n t h i s la t e r) ,

t h e o p t i m a l v a l u e o f ( P~ ) i s g iv e n b y

Z * = m i n Z * ( n l ) . ( 8 )n l ~ ( 1 , N )

W i t h n 1 j o b s a s s i g n e d t o d u e - d a t e 1 , w e f i n d Z * ( n 1 ) b y c o n s i d e r i n g t h e t w o c a s e s a s d i s c u s s e d e a r l i e r .

W e n o w a n a l y z e t h e s e t w o c a s es s e p a r a t e l y i n o r d e r t o o b t a i n a l o w e r b o u n d a n d a n u p p e r b o u n d t o (P ~).C a s e I : Ctn ,l = ~ -, n 1 f i xe d . C on s t r a in t s o f ( P2) r e d uc e t o

n I

E t[i] - < " r , ( 9 )

i = 1

N

~ , t tq < r . ( 1 0 )

i = n l + l

n I tl 1AS Y~i=nl+lt[ ilN = T - ET L 1 t i l l, ( 10 ) b e c o m e s , T - ~ , i~ i t[ i] - r o r ~ , i = l /[ il >- T - r . W i th t he a bo ve ob se r va -

t i o n s , a n e q u i v a l e n t f o r m u l a t i o n o f ( P 2 ) w i t h a n a d d i t i o n a l r e q u i r e m e n t t h a t n ~ i s f i x e d c a n b e w r i t t e n a s:

( P 4 ) n l N - - n 1

m i n Z l ( n l ) = ~'~ ( i - l ) t [ i l + E ( i - 1 ) t[ i+ n l l+ ( 2 N - - n l ) r 6o ,~ 1 2 i = 1 i = 1

n l

s . t . ~ , t t i ] < " r , ( 1 1 )i = 1

n l

- - E t[ i] <- "r - - T , (12 )i = 1



D. Chhajed / European Journal of O perat ional Research 84 (1995) 385-401 391

Note that the last term in the objective function is a constant for a fixed n 1. We form a Lagrangian

relaxation of (P4) by multiplying constraints (11) and (12) by non-negative numbers "~1 and h 2,

respectively, and adding them to the objective function:

n I N - n 1

mi nZ l( nl , hi, h2) = ~ (h I - h 2 + i - 1)t[i]+ ~ ( i - 1)t[i+n,]

o , ~ /2 i = 1 i = 1

+ (2N - n , ) 7 - 6 - h,7- + Az( T- 7-). (13)

The choice of (hi, h2) that solves

_Z~'(nl) = max Z~' (n l, h,, h2) (14)/~I,A2

provides us with the best choice of the Lagrangian multipliers in the sense of providing the tightest lower

bound. Thus,

_Z~(nl) = maxZ~'(n l, /~1, h2) < Z ~( n l ) .A I , A 2

We note that for fixed values of n x, hx and h 2, the first two terms o f (13) are the multiplication of two

series. The first series consists of the processing times while the second series is formed by the positional

weights. To minimize the product o f two series we arrange them in reverse order, i.e., if ti's are ar ranged

in non-decreasing order then the positional weights are arranged in non-increasing order. The rank of a

positional weight corresponds to the index of the job assigned to that position.

C a s e I I : C [ n l ] < 7-, n x fixed. In order to have a schedule with the completion t ime of the last job

assigned to the first due-date not coincide with 7-, we must have

n l

E t[ i ] < 7 ", ( 1 5 )

i = 1

N

E tie ] > 7 -. ( 1 6 )

i = n l + l

Note that the start time of the first job for this case is 27--T. Constrain t (15) restricts the total processing

times of jobs assigned to due-date 1 to less than the period and constraint (16) is required because the

total processing time of jobs assigned to due-date 2 must exceed 7-.• g _ n I n 1

With ~ _ , i = n ~ + l t [ i ] - T - Y ~ 3 = l t [ i ] , (16) becomes T - ~ , T L l tt q > 7- or T - 7- > Y',i=lttq. Since T - 7- < 7-,constraint (16) implies (15). (P3) with n I fixed can be rewritten as:

N

min Z2(n,) = Y ~ ( i - 1 ) tt i ] + (2 N- n,)zt$ -n ,zo ' E . Q i = 1

s.t. ~ t ti I < T - r . (17)i = 1

The last two terms in the objective function are constant. Again we form a Lagrangian relaxation bymultiplying constrain t (17) by A and including it in the objective function to get

n l N

minZ2(nl , A)= ~(A +i- 1) tv l + ~ ( i - 1 ) t H + ( 2 N - n l ) 7 - 6 - n l , - ( T - , ) h . (18)t rE g 2 i = 1 i = n l + l



3 9 2 D. C hhajed European Journal of Operational Research 84 (1995) 385-401

W e w a n t a v a l u e o f * t h a t s o l v e s t h e d u a l p r o b l e m

Z ~ ' ( n l ) = m a x Z ~ ' ( n l , * ) < Z ~ ' ( n l ) .A

T h e o r e m 1 s h ow s h o w a l o w e r b o u n d t o 2 D S P c a n b e c o m p u t e d .

T h e o r e m 1. The va lue o f

( m i n { Z ~ ' ( n , ) }

Z * = ~ n ~ -

- / m i n ( m i n { - - Z ~ ' ( n l ) } , m i n { - Z ~ ' ( n l ) } ) ,x n I n I

i s a va l id lower bound to ( 2 D S P ) .

i f 2z - T > t N

otherwise ,

( 1 9 )

Proof . T h e p r o o f f o ll o w s e a s il y f r o m t h e a b o v e d e v e l o p m e n t , rE .

T h e d u a l p r o b l e m s ( 1 4) a n d ( 1 9 ) c a n b e s o l v ed b y a n y o f t h e m e t h o d s g i v e n in F i s h e r ( 19 8 1 ). W e n o w

d e v e l o p s e v e r a l r e s u lt s t h a t r e d u c e t h e c o m p u t a t i o n a l t i m e r e q u i r e d t o o b t a i n _ Z* a n d p r e s e n t a n

a l g o r i t h m t h a t u s e s t h e s e r e s u l t s t o c o m p u t e Z * .

L e m m a 2 . There exists an optimal solution to ( 14) with a t least one o f *1 and *2 equal to O.

Proof . W e p r o v e t h i s b y c o n t r a d i c t i o n . S u p p o s e f o r f ix e d n l , w e h a v e m u l t i p l i e r s * * > 0 a n d * '~ > 0 f o r

w h i c h Z T ( n l , * .1 , * ~ ) i s a n o p t i m a l s o l u t i o n t o (14) .

Case A: *.1 > *~ . I f w e use ne w m ul t i p l i e r s X1 = *.1 - *~ an d * '2 = *~ - *~ = 0 , th e re la t iv e ra nk ing of

t h e p o s i t io n a l w e i g h t s d o e s n o t c h a n g e a n d s o t h e g i v e n s e q u e n c e is st il l o p t i m a l w i t h t h e s e n e w v a l u e s o f

t h e m u l t i p l ie r s . T h e n e w o b j e c t i v e f u n c t i o n v a l u e i n ( 1 3 ) i sn I N - n 1

Z ~ ( n , , X 1, X 2) = ~'~ ( X 1 - X 2 + i - 1 ) t t i l + • ( i - 1 ) t [ i + n d + ( 2 N - n l ) r t S - X l T - X 2 ( ¢ - T )i = 1 i = 1

* ' 1 , * * 2 ) r ) .

A s 2 ¢ > T , Z ' ~ ( n l , *'1, A'2) > Z ~ ( n l , A*1, A ~ ) , w e h a v e f o u n d a s o l u t i o n w h i c h i s a t l e a s t a s g o o d a s t h e

g i v e n o p t i m a l s o l u t i o n t o ( 1 4 ) a n d s a t i s fi e s th e l e m m a .

Case B: A*1 < A ~. T h e p r o of o f t h i s c a se i s s imi l a r t o t h e p r o of o f C a se A . N o w w e se t A '1 = A*1 - A~ a n d, ' 2 [ ]

L e m m a 3 . It is suf f icient to consider only integer values o f the multipliers in ( 13) a n d (18) .

Proof . C o n s i d e r ( 1 3 ) . O n l y o n e A ( i = 1, 2 ) is p o s i ti v e . S u p p o s e *1 > 0 a n d k < A 1 < k + 1 f o r s o m e

i n t e g e r k . F o r a n y v a lu e o f A 1, Z ~ ( n 1, A1, 0 ) i s o b t a i n e d b y a r r a n g i n g t h e s e q u e n c e { (* 1 + i -

1 ) : 1 . . . . n l } u { (i - 1 - n l ) : i = n 1 + 1 , . . . , N } i n n o n - i n c r e a s i n g o r d e r a n d m u l t i p l y in g i t w i th t i 'S. I f w e

v a r y A~ b e t w e e n k a n d k + 1 , t h e o r d e r i n g o f th e s e q u e n c e d o e s n o t c h a n g e a n d s o th e f i r st tw o t e r m s i n

( 1 3 ) a r e l i n e a r f u n c t i o n s o f A~. T h u s ( 1 3 ) , w i t h k < h I < A l < k + 1 h a s a n o p t i m a l s o l u t i o n a t a n e x t r e m e

po int , i .e . , A~ - k or A 1 = k + 1.

T h e p r o o f s o f i n t e g r a l i t i e s o f A 2 i n ( 1 3 ) a n d A i n ( 1 8 ) a r e s i m i la r . [ ]



D. Chhajed / European Journal of O perational Research 84 (1995 ) 385-401 393

B y th e v i r t u e o f t h e a b o v e t w o l e m m a s , w e o n l y h a v e t o c o n s i d e r i n t e g e r v a l u e s o f t h e m u l t ip l i e r s.

F u r t h e r m o r e , w h i le c o m p u t i n g Z ~ ( n 1, A1, A 2 ) , a t m o s t o n e o f th e t w o m u l t i p l ie r s w il l b e n o n - z e r o . W e

n e e d t h e f o l lo w i n g d e f i n it i o n s f o r t h e n e x t t h e o r e m .

L e t :1

m 0 = m i n i m u m i n t e g e r >_ ~ N .

m I = m a x { j : ~ / = l t i ~ ~'}.= ~ i= l tN _ i+ l <__ T - r } .2 m i n{ j : i

m 3 = m a x { j : Y' ./=lt i < T - ~}.

N o t e t h a t i n d e f i n i n g m a a n d m 3 w e a d d t h e s m a l l e s t j o b s , w h e r e a s i n d e f in i n g m 2 w e a r e a d d i n g t h e

l a r g e s t j o b s . T h e o r e m 2 p r o v i d e s v a l u e s o f n ~ a n d t h e m u l t i p li e r s (A 1, A2 , A) t ha t a r e s u f f i c i e n t t o

e v a l u a t e Z * .

T h e o r e m 2 .

a ) F o r

a l )

a 2 )

a 3 )b ) F o r

b l )

b 2 )

T h e l o w e r b o u n d Z _* c a n b e c o m p u t e d b y r e s tr i c ti n g t h e r a n g e o f s e a r c h t o :

Z l ( n l , A I , A 2 ):

m ax {m E, m 0} _< n I _< m l ,

A 1 ~ ( 0 , N - n x ) , a n d

A2 ~ (0 , n I - 1) .Z 2 ( n l , A ) :

N - m I - 1 <_ n 1 < _ m 3 a n d

A ~ ( O , N - l ) .

P r o o f . T h e P r o o f o f t h is t h e o r e m is g i v e n i n A p p e n d i x B . [ ]

T h e p o s i t i o n a l w e i g h t s o f fi r st n ~ p o s i ti o n s a r e a l l d i f f e r e n t. H o w e v e r , o n e o r m o r e p o s i t i o n a l w e i g h t s

o f th e f i r s t n 1 p o s i t io n s m a y b e e q u a l t o th e l a t t e r N - n I p o s i ti o n a l w e i g h ts . S u p p o s e t w o p o s i t io n a l

w e i g h t s Wp a n d w q a r e e q ua l w i t h p _< n 1, q > n~ a nd l e t i a n d i + 1 be t he r a n ks o f Wp a n d w q . I n a n y

o p t i m a l s o l u t i o n t o t h e r e l a x e d p r o b l e m s (1 3 ) a n d ( 1 4 ), j o b s i a n d i + 1 w i ll b e a s s i g n e d t o p o s i t i o n s p

a n d q a n d t h i s a s s i g n m e n t c a n b e d o n e i n t w o w a y s. W e , h o w e v e r , fo l lo w t h e c o n v e n t i o n o f a s s i g n in g t h es m a l l e r j o b ( i .e . j o b i ) to p o s i t i o n p t h e r e b y a s s i g n i n g i t t o t h e f i r s t d u e - d a t e . F o r a g i v e n v a l u e o f n 1 a n d

t h e m u l t ip l i e rs , l e t F b e t h e s e t o f j o b s a s s i g n e d t o t h e f i r s t d u e - d a t e , S b e t h e s e t o f j o b s a s s i g n e d t o t h e

s e c o n d d u e - d a t e a n d l e t t ( a ) b e t h e s u m o f t h e p r o c e s s i n g t i m e s o f j o b s i n th e s e t o f a .

F o r a g iv e n v a l u e o f n ~ , w e n o w g i v e n e c e s s a r y a n d s u f f ic i e n t c o n d i t i o n s t o d e t e r m i n e t h e o p t i m a l

v a l u e o f th e m u l t i p l ie r s h a n d ( A 1 , A 2 ) i n ( 1 9 ) a n d ( 1 4 ) , r e s p e c t i v e l y . W e f i r s t a n a l y z e C a s e I I .

T h e o r e m 3 . F o r a f i x e d v a l u e o f n 1, t h e o p t i m a l v a l u e o f A t h a t s o l v e s ( 19 ) is A* = min{A : t ( F ) <_ T - r} ,

w h e re t ( F ) = El= 1 . . .. . l t [ i ] •

P r o o f . T h e P r o o f i s g i v en in A p p e n d i x C .

T h e o r e m 4 . F o r a f i x e d v a l u e o f n l , t h e o p t i m a l v a l u e o f A 1 a n d A2 t h a t s o l v e s ( 14 ) a r e A] = min{Al : t ( F )

< ~ '} and A* = m a x{Az : t ( F ) > T - ~'}.

P r o o f. T h e P r o o f i s s i m il a r to t h e P r o o f o f T h e o r e m 3 .

F r o m t h e P r o o f o f T h e o r e m 2 in A p p e n d i x B , n o te t h a t w h e n h o r A 1 = N - nl , t ( F ) <_ r a n d w h e n

h 2 = n 1 - 1, t ( F ) >_ T - r . T h u s t h e r e e x i st s o p t i m a l m u l t i p l i e r s t h a t w il l s i m u l t a n e o u s l y s a t is f y T h e o r e m

2 , a n d T h e o r e m s 3 o r 4. A l s o , w i t h t h e o p t i m a l v a l u e o f A 1 >__ 0 ( A 2 = 0 ) i f t ( S ) <_ z t h e n t h e o p t i m a l



3 9 4 D. Chhajed European Journal of Operational Research 84 (1995) 385-401

a s s i g n m e n t o f j o b s a l s o g i v e s a f e a s i b l e ( C a s e I ) s o l u t i o n t o ( P 1) . S im i l a r ly , w i t h / ~ 1 - ~ " 0 an d A2 = , t~ > 0 , i f

t (F ) < ~" t h e n t h e o p t i m a l s o l u t i o n t o ( P 4 ) i s a l s o a C a s e I f e a s i b l e s o l u t i o n t o ( P1 ). W h i l e s o l v i n g ( 1 9 ) w i t h

A*, if t ( S ) > z t h e n t h e s o l u t i o n i s a C a s e I I f e a s i b l e s o l u t i o n o t h e r w i s e i t i s a C a s e I f e a s i b l e s o l u t i o n .

T h e c o s t o f a n y o f t h e s e f e a s ib l e s o l u ti o n s i s a n u p p e r b o u n d t o (P 1). T h e r e f o r e , w h i l e g e n e r a t i n g t h e

l o w e r b o u n d t o ( P 1 ) , u p p e r b o u n d s c a n a l s o b e c o n s t r u c t e d .

T h e p r o c e s s o f o b t a i n i n g t h e a s s i g n m e n t o f jo b s , g i v e n n 1 a n d t h e m u l t i p li e r( s ) , c a n b e f u r t h e rs i m p l i f i e d . T o s e e t h i s , c o n s i d e r ( 1 3 ) . W h e n A 2 = 0 a n d A~ > 0 , t h e p o s i t i o n a l w e i g h t s o f t h e j o b s a s s i g n e d

t o t h e f i r st a n d t h e s e c o n d d u e - d a t e s a r e { A 1, A1 + 1 , . . . , A1 + n l - 1} a nd { 0, 1, 2 . . . . N - n l } , r e spe c -

t iv e ly . F o l l o w i n g t h e c o n v e n t i o n o f a s s i g n i n g t h e s m a l l e r j o b t o F i n t h e c a s e o f a ti e , t h e o p t i m a l

a s s i g n m e n t c a n b e s t a t e d a s a r u l e g i v e n b e l o w ( R 1 ) . I n t h i s r u l e a n d t h e t w o a d d i t i o n a l r u l e s t h a t f o l l o w ,

w e a s s i g n j o b s f r o m t h e l a r g e s t t o t h e s m a l l es t .

R 1 . A s s i g n t h e l a r g e s t A 1 j o b s ( J N , . . . , J N - ~ + 1 ) t o S . S t a r t i n g w i t h j o b J N - a~, a l t e r n a t e b e t w e e n S a n d F

u n t i l a t le a s t o n e o f S o r F i s ful l . A s s i g n t h e r e m a i n i n g j o b s t o t h e n o n - f u l l d u e - d a t e . H e r e , w e s a y F is

fu l l i f i t ha s n I j obs a nd S i s fu l l i f i t ha s N - n I j obs .

F o l l o w i n g a r g u m e n t s s i m i l a r to a b o v e , t h e r u l e f o r a ss i g n i n g j o b s i n ( 1 3 ) w h e n / ~1 = 0 , ~ 2 ~ -- 0 is:

R E . A s s i g n t h e l a r g e s t A2 j o b s t o F . S t a r t i n g w i t h JN-~2' a l t e r n a t e l y a s s i g n j o b s t o S a n d F u n t i l a t l e a s t

o n e o f t h e m i s f u ll . A s s i g n t h e r e m a i n i n g j o b s t o t h e n o n - f u l l d u e - d a t e .

R u l e R 3 i s f o r f i n d i n g a n o p t i m a l a s s i g n m e n t o f j o b s i n ( 1 8 ).

1 13 . L e t k = m a x ( n 1 - A , 0 ) . A s s i g n t h e l a r g e s t k j o b s t o F . S t a r t i n g w i t h J N - k , a l t e rn a t e b e t w e e n S a n d

F u n t il o n e o f t h e m i s f u ll a n d t h e r e a f t e r a s si g n th e r e m a i n i n g j o b s t o t h e n o n - f u l l d u e - d a t e .

W e n o w o u t l i n e t h e s o l u ti o n p r o c e d u r e t o o b t a i n t h e l o w e r b o u n d _Z* a n d a n u p p e r b o u n d ( U B ) to

2 D S P .

Algorithm DD BOUN DSI n t h e a l g o r it h m , U B i ( n 1 ) i s d e f i n e d t o b e a v e r y la r g e n u m b e r f o r a n i n f e a s ib l e s o l u ti o n .

Step 1. {Find Z~ ' (n 1 , A , 0 ) fo r ma x{m E, m 0} _< n I < ml .}

F o r / 71 = ma x{ m E , m 0} t o m I do

Step 1.1. S e t A = 0 , / ~ 2 = 0 .

Step 1.2. U s e R 1 t o f i n d F a n d S . C o m p u t e t ( F ) .

Step 1.3. I f t( F ) <_ ~ t h e n c o m p u t e Z ~ ( n l , A 1, 0 ) , s e t A ] = A1 a n d g o t o S t e p 1 .4 .

E l s e s e t ) t 1 = A 1 + 1 ; G o t o S t e p 1 .2 .

Step 1.4. S e t U B 1 ( n 1) t o t h e v a l u e o f t h e s o l u t i o n f o u n d i n S t e p 1 .2 .

Step 2. { F i nd Z ~ (n 1, O, AE) fo r m ax { m z , m 0} _< n 1 < m l.}

F o r n 1 - - m a x { m E, m o} t o m 1 d oStep 2 .1 . Se t A1 = 0, A2 = 1.

Step 2 .2 . U s e R 2 t o f i n d F a n d S . C o m p u t e t ( F ) .

Step 2.3. I f t ( F ) >_ T - • t h e n c o m p u t e Z ~ ( n j , 0, A 2); se t A~ = ~ 2 a n d g o t o S t e p 2 . 4 .

E l s e s e t Az = A2 + 1 ; G o t o S t e p 2 .2 .

Step 2 .4 . S e t U B E ( n 1 ) t o t h e v a l u e o f t h e s o l u t i o n f o u n d i n S t e p 2 . 2 .

Step 3. Set _Z~'(n1) = m ax {Z *( n 1 , A~, 0) , Z ~ (n 1, O, A~ )} fo r n 1 ~ (ma x{ m E, m0}, m l ) .

Step 4. { F i nd _Z~ (n 1 ) i f Le m m a 1 is no t s a t is f i e d .}

If 2~" - T _> t N t he n se t U B3 (n 1) an d _Z~ '(n1 ) t o a v e r y l a r g e n u m b e r a n d g o t o S t e p 5 .



D. C hhajed European Journal of Operational Research 84 (1995) 385-401 395

S tep 5 .

Step 6 .

Otherwise, for n 1 = N - m 1 to m 3 do

S t e p 4 . 1 . Set A =0 .

S tep 4 . 2 . Use R3 to find F and S. Compute t ( F ) .

Step 4 .3 . If t ( F ) <_ T - ~ then compute _Z~'(n~) = Z ~ ( n l , A) and go to Step 4.4.

Else set A = A + 1; Go to Step 4.2.

S tep 4 . 4 . Set UB3(r/1) to the value of the solution found in Step 4.2.Set _Z* = min(minn~{_Z*(nl)} minnl{_Z~'(nl)}).

Set UB = minnlmin{UBl(nl), UBE(nl), UBa(nl)}}.

To obtain the time complexity, note that using rules R1, R2 and R3 to find sets F and S takes O(N)

time. t ( F ) and Z ~ ( n l , A1, 0) each can be computed in O( N) time. Since there are O(N) possible values

of the multiplier A1, Steps 1.1-1.4 can be computed in O(N 2) time. This is repeat ed for O(N) values of

rtl, resulting in O(N 3) time complexity for Step 1. The same analysis is true for Steps 2 and 4. The initial

sorting of the processing time takes O(N log N) time. Thus the complexity of the algorithm as stated is

O(N3). The time complexity can be further reduced by noting that the functions Z ~ ( n t , A 1, 0 ), Z * ( n l , O ,

A2), and Z ~ ( n l , A) are piece-wise concave functions of the multipliers (Nemhauser and Wolsey, 1988).

Thus for a given value of n 1, if we perform a binary search over A1, instead of a linear search as in theabove algorithm, only O(log N) computations of Z ~ ( n ~ , A1, 0), will be required. Similarly for Z ~ ( n l , O ,

, ~ . 2 and Z ~ ( n l , A ) . This will reduce the time complexity to O(N 2 log N).

4 . Emp ir ica l an a lys i s

To investigate the effectiveness of the proposed lower and upper bounds we coded the heuristic

algorithm in the Pascal programming language on a personal computer (Macintosh IIcx). A 34 experi-

mental design was used involving number of jobs (N); processing times of jobs; tightness of due-date;

and the due-da te penalty. The values of N were (20, 40, 80). The processing times (integer) of the jobs

were generated from a uniform distribution between (1, tmax) with tmax being (10, 20, 30). The tightnessof the due-da te refers to the relative amount of idle time. We set the time period z = a S ' t i where a is a

parameter with values (1.1, 1.3, 1.5). Here a = 1.1 implies 10% idle time in the data. The earliness

penalty was set to 1 and (0.1,.75, 1.25) were the values used for the due-da te penalty per job per time

period. Thus 81 different problems were studied. For each problem five replications were done. Lower

and upper bounds were computed using the algorithm DDBOUNDS.

Table 1 summarizes the average and the maximum values of the relative gap, measured as 100 * (upper

bound- lower bound)/lower bound. As Table 1 indicates, the lower and the upper bounds are very

close. The maximum gap for the 405 problems we solved was 0.24%. For over 65% of the problems,

optimal solution was found. The maximum time for the 80 job problems was approximately 13 seconds on

Macintosh IIcx, not including the inpu t/o utput time.

5 . Exten s ion s an d con c lu s ion

This paper introduces a new scheduling problem with due-date and earliness costs that has applica-

tions in JIT environments. The problem with two due-dates was shown to be NP-hard. The unrestricted

case was shown to be easily solvable by using a procedure similar to the procedure given by Kanet (1981)

and a lower bound and upper bound for the restricted case were provided. Computational experience

suggests that these bounds are quite tight.



396

Table 1

D. Chha jed European Journal o f Operational Research 84 (1995 ) 385-4 01

tmax a = 1.1 a = 1.3 a = 1.5

= 0.1 6 = 0.75 t~ = 1.25 6 = 0.1 6 = 0.75 t~ = 1.25 ~ = 0.1 ~ = 0.75 ~ = 1.25

N = 20 10 0 a 0.076 0.052 0 0 0 0 0 00 b 0.24 0.16 0 0 0 0 0 0

20 0 0.13 0.084 0 0.032 0.02 0 0 00 0.21 0.13 0 0.11 0.07 0 0 0

30 0.002 0.024 0.031 0.012 0 0.016 0.01 0 0.011

0.011 0.12 0.07 0.03 0 0.08 0.05 0 0.038

N = 40 10 0 0 0 0 0.026 0.018 0 0.006 0.004

0 0 0 0 0.07 0.05 0 0.03 0.0220 0 0.049 0.035 0 0.016 0.01 0 0 0

0 0.14 0.1 0 0.03 0.02 0 0 0

30 0 0.07 0.044 0 0.008 0.006 0 0 0

0 0.122 0.08 0 0.02 0.01 0 0 0N = 80 10 0 0.001 0 0 0 0.0 0 0.0 0

0 0.01 0 0 0 0 0 0 0

20 0 0.006 0.004 0 0.009 0.006 0 0.005 0.005

0 0.032 0.022 0 0.017 0.011 0 0.008 0.014

30 0 0.007 0.005 0 0.012 0.006 0 0 0.0010 0.02 0.012 0 0.029 0.02 0 0 0.006

gap = 100, (ub - Ib)/ lb.

a First number: average gap of 5 replications.b Second number: maximum gap of 5 replications.

The algori thm outl ined above can be modified for the case when the two periods are not of equal

length. Su ppose the first peri od has le ngth ~l and the secon d pe riod 's le ngth is ~'2 with z 1 + 72 > T. In

this case it is easy to see that z in the right ha nd side (rhs) of (11) and (15) should be c hang ed to zl, ~- in

the rhs of (12) an d (16) shoul d be c ha ng ed to z 2. As a res ult, th e rhs of (17) will be T - ~'2. Th e rule s R1,

R2 an d R3 will st i ll remain valid but the ranges of mult ipliers in Theo rem 2 and the definit ions of

m 0 .. . . m 3 will change. We do no t re port these chan ges as they can be derived by following the earlier

development.

The method provided above can be used as a heurist ic to develop good feasible solutions to the

general problem with more than two due-dates. We discuss two possible uses: (i) to develop feasible

solution and (ii) to improve feasible solution. To develop a feasible solution suppose we have a problem

with N jobs and 4 due-d ates at intervals of 1 time unit. We first create a probl em with the same N jobs

but with 2 due -dat es at intervals of 2 time unit. For each feasible value of n I find the 'be st ' (Case I or

Case II) part i t io ning of jobs, F ( n l ) and S ( n l ) , between 'due-date 1 ' and 'due-date 2 ' . Note that in

comput ing F ( n 1) and S ( n l ) we do not need the due-date penalty. For the jobs in F ( n l ) , we again solve a

2 due-date probl em with appropriately defined t ime intervals and the same is done for the jobs in S ( n ~ ) .

Ap pe nd in g these two solutions gives a solution to the ori ginal probl em with n I jobs assigned to the first

two due-dates and we compute the cost of this solution. Repeat this process for every feasible value of na

and select the best feasible solution.

The methodology developed in previous sections can also be used to improve a given feasible solution.

We select two due dates and jobs assigned to them and reassign these jobs to the two due dates using the

above scheme if it results in lower objective funct ion value. This is analogo us to the pair-wise

impro vemen t schemes, except that here we are using a pair of due-dates for reas signmen t of the jobs.

There are other possible extensions of the problem, some of which are subject of our on-going

research. These include

(i) iden tifa tion of special cases that are solvable in polyn omial time;



D. Chhajed / European Journal of O perational Research 84 (1995 ) 385-401 3 9 7

( i i) v a r i a b l e d e l i v e r y i n t e r v a l s a n d t h e p o s s i b i li t y o f d i s p a t c h i n g a d d i t i o n a l t r u c k s w i t h e x t r a c o s t o r

p e r h a p s n o t m a k i n g a t r i p i n s o m e t i m e p e r i o d s ,

( ii i) t r u c k c a p a c i t y c o n s t ra i n t s , j o b d e p e n d e n t e a r l in e s s p e n a l t i e s, a n d d i f f e r e n t r e l e a s e t i m e s o f j o b s .

A c k n o w l e d g m e n t

I w o u l d l i k e to t h a n k C h u n g - Y e e L e e o f th e U n i v e r s it y o f F l o r i d a f o r p r o v i d i n g m e w i t h L e m m a 1 .

A p p e n d i x A

P r o b l e m c o m p l e x it y

W e s h o w t h at 2 D S P is N P - h a r d b y r e d u c in g t h e N P - h a r d e v e n - o d d p a r ti t io n i n g p r o b l e m ( G a r e y ,

T a r j a n a n d W i l fo n g , 1 9 8 8 ) t o 2 D S P i n p o l y n o m i a l ti m e .

E v e n - O d d P a r t i t i o n i n g P r o b l e m ( E O P P ) . G i v e n 2 n n u m b e r s, 0 < x 1 <x2 . . . < x 2n , doe s t he re e x i s t a

p a r t i t i o n X 1 an d X 2 o f t he se num be rs su c h t ha t ~ 'x ~~ x~Xi = Ex~ ~ x 2 X i a n d e x a c t l y o n e o f { X 2 i _ 1 ' X2i) is in

X 1 ( a n d s o i n X 2 ) f o r i = 1 . . . . n .

G i v e n a n i n s t a n c e o f E O P P , w e d e f i n e a n i n s t a n c e o f 2 D S P w i t h 2 n j o b s . T h e p r o c e s s i n g t im e o f t h e1_ z n l v . 2 n t = ( n + 7 ) p . .j o b s a r e t i = x i + ~ , f o r i = l , . . . , 2 n , w i th / x - E ~ = l X i a n d t h e t im e p e r io d 7- = ~ z . . . i= l i

O b s e r v e t h a t t 1 < t 2 • • • < t z , a n d t h e t o t a l p r o c e s s i n g t i m e o f t h e 2 n j o b s i s ~ . i t i = 2 7 " . S o , i t i s pos s i b l e

t o c o m p l e t e a l l t h e jo b s i n t h e f ir s t t w o t i m e p e r i o d s a n d n o i d l e t i m e w i ll o c c u r . L e t t h e e a r l in e s s p e n a l t y

b e 1 a n d t h e d u e - d a t e p e n a l t y 8 = 2 " .

C o n s i d e r t h e d e c i s i o n v e r s i o n o f 2 D S P : F i n d a s c h e d u l e w i th c o s t < O = 3 n ~ ' 2 " + E n = l ( i - 1 )

( t 2 ~ , - i + l ) + t 2 t , - 1 ) + 1 ), i f i t e x is t s. W e s h o w t h a t s o l v in g t h i s d e c i s i o n p r o b l e m g i v e s u s a s o l u t i o n t o t h e

E O P P , t h u s s ho w i n g t h a t 2 D S P is N P - h a r d .

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

d u e - d a t e . I t i s p o s s i b l e t o c o m p l e t e n j o b s ( a s E T = l t i < 7") i n t h e f i r st t i m e - i n t e r v a l , b u t i t is n o t p o s s i b l e

t o c o m p l e t e m o r e t h a n n j o b s i n th e f i r s t p e r i o d s i n ce t h e s u m o f t h e p r o c e s s i n g t i m e s o f t h e s m a l l e s t

n + 1 jo bs is E ~ + 1 ~-'n + 1= 1 t i = ( n + 1) / .~ + ~-- , i= 1 x i > 7". T h e r e f o r e , t h e o p t i m a l s c h e d u l e w i l l h a v e e x a c t l y n j o b s

a s s i g n e d t o e a c h o f t w o d u e - d a t e s . W e n o w a n a l y z e t h e t w o c a s e s c o n s i d e r e d i n S e c ti o n 2 w i t h t h e v a l u e

o f n 1 f i x e d t o n .

Case I : T h e c o m p l e t i o n t i m e o f t h e n - t h j o b c o i n c id e s w i th r . T h e t o t a l p e n a l t y o f s u c h a s c h e d u l e i s

g i v e n b y ( 5 ), w h i c h i n t h i s c a s e , i s

n

Z l ( n ) = ~ ( i - 1)( t [ i I + t tn+i l ) + 3nT"tS .i= 1

T h e o p t i m i za t io n p r o b l e m c a n b e w r i t te n a s

m i n Z l ( n )

s . t . C M = 7". (A .1 )

L e t Z ~ ( n ) t h e o p t i m u m s o l u t i o n v a l u e to t h e a b o v e p r o b l e m a n d l e t Z ~ ' ( n ) b e th e o p t i m u m v a l u e t o t h e

a b o v e p r o b l e m w i t h c o n s t r a i n t ( A . 1 ) r e l a x e d t o C M < r .



398 D. Chhajed / European Journal of O perat ional Research 84 (1995) 385-401

Case I I : C tn ] < r . T o c o m p u t e t h e t o t a l c o s t o f s u c h a s c h e d u l e c o n s i d e r t h e o b j e c t iv e f u n c t i o n o f ( P 0 ,

2n 2n

E ( 'T - - C [ / ] ) + E ( 2 7 " - C [ i ] ) + 3n~ '6

i=1 i - n + l

n 2n

= E ( C t n ] - C t i ] ) + E ( C [ 2 n ]- - C [ i ]) + 3 n r 6 + n ( r - C [ n ] )i=1 i=n+l

= ~ ( i - 1 ) ( tt i I + / t n + i l ) + 3 n ~'6 + n ( ~ - - C M ) -- Z 2 ( n ) . ( m . 2 )i=1

T h u s , t h e o p t i m i z a t i o n p r o b l e m f o r C a s e I I i s :

m i n Z 2 ( n )

s.t . Ct~ 1 < ~-. (A .3 )

L e t Z ~ ' ( n ) b e t h e m i n i m u m v a l u e o f Z 2 ( n ) w i t h (A . 3 ) re l a x e d t o C M < r . S i n c e t h e f ir s t t w o t e r m s o f

_ Z ~'( n) a r e m i n i m i z e d b y _ Z ~ (n ) a n d t h e t h i r d t e r m o f Z 2 ( n ) i s n o n - n e g a t i v e , _ Z~ '( n) ~< Z ~ ' ( n ) . N o t e t h a t

Z ~ ( n ) i s m u l t i p l i c a ti o n o f t w o s e r ie s o f 2 n n u m b e r s . O n e s e r ie s i s fo r m e d b y t h e p r o c e s si n g t i m e s o f t h e

2 n j o b s a n d t h e o t h e r s e r i e s co n s is t s o f in t e g e r s { 0 , . . . , n - 1} w i t h e a c h i n t e g e r r e p e a t e d t w i c e . T o

m i n i m i z e th i s s u m o f p r o d u c t s ( w i t h o u t c o n s t r a i n t ( A . 1 )) , w e h a v e t o a r r a n g e o n e s e r i e s i n i n c r e a s i n g

o r d e r a n d t h e o t h e r i n d e c r e a s i n g o r d e r . T h i s i s e q u i v a l e n t t o a s s i g n i n g j o b s { 1, 2} t o p o s i t i o n { n , 2 n }, j o b s

{3, 4} to po si t io ns {n - 1 , 2n - 1} , . . . , an d job s {2n - 1 , n} to po si t io ns {1, n + 1} . Th is g ives

n

3n~'2~" + E ( i - 1 ) ( t 2 ( n _ i + l ) + / 2 ( n - 1 ) + l ) = ~ _< Z~ (n ).

i= 1

I f s u c h a n a s s i g n m e n t a l s o sa t i s fi e s ( A . 1 ) t h e n t h e r e s u l t i n g s c h e d u l e i s o p t i m a l C a s e I s c h e d u l e .n m. ~2 n t =E q u a t i o n ( A . 1) i s s a t is f ie d i f a n d o n l y i f t h is a s s i g n m e n t c a n b e m a d e s u c h t h a t E i = l t t q z.,i=n+1 [i] 7.

S i n c e t[i ] = a[i ] + Iz, s u c h a n a s s i g n m e n t i s p o s s i b l e i f a n d o n l y i f E O P P h a s a s o l u t i o n .

T h e a b o v e s c h e d u l e i s s u c h t h a t _ Z ~ ( n ) = Z ~ ( n ) = Z _~ (n )< _ Z ~ ( n ) . T h u s i t i s t h e o p t i m a l s o l u t i o n t o

t h e s c h e d u l i n g p ro b l e m .

A p p e n d i x B

P r o o f o f T h e o r e m 2 . a l ) F o r C a s e I , i t i s c l e a r t h a t a t l e a s t a s m a n y jo b s a r e a s s i g n e d t o t h e f i r s t d u e - d a t e

a s i n t h e s e c o n d . ( O t h e r w i s e i n t e r c h a n g i n g t h e d u e - d a t e s o f a l l j o b s g i v es a b e t t e r s o l u t i o n . ) T h u s n 1

s h o u l d b e a t l e a s t e q u a l t o m 0. W e n o t e t h a t m 2 is t h e m i n i m u m n u m b e r o f j o b s r e q u i r e d t o s a t is f y ( 12 )

a n d s o n 1 c a n n o t b e s m a l l e r t h a n m E. T h e r e f o r e , n I h a s t o b e a t l e a s t t h e m a x i m u m o f m z a n d m 0.

S i n c e t h e s u m o f t h e s m a l l e s t m I + 1 j o b s e x c e e d s ~- a n d t h u s v i o l a t e s ( 1 1 ), n 1 c a n n o t b e g r e a t e r t h a n

m 1

a2 ) B y Le m m a 2 , i f A~ > 0 , A2 = 0 . B y L e m m a 3 w e r e s t r i c t A 1 tO n o n - n e g a t i v e i n t e g e r s . C o n s i d e r ( 1 3 )

a n d l e t t h e p o s i t i o n a l w e i g h t s b e

w i = A l + i - - 1 , i = 1 . . . . . n l , u i = i - 1 , i = 1 . . . . . N - n 1 .

N o t e t h a t W l ~ . W 2 < " '" < W n l a n d U I < U 2 < " '" < U N _ n • I f h ~ > ~ N - n ~ , U N _ n I < _ ~ W I a n d t h u s t h e

s m a l l e s t n I j o b s a r e a s s i g n e d t o t h e f i r st d u e - d a t e t o g e t Z ~ ( n~ , h~ , O ). T h i s o p t i m a l v a l u e i s e q u a l t o

Z ~ ( n l , A l , 0 ) = ~ w i G l + l _ i + ~ v i t N + l _ i + ( 2 N - n x ) ~ ' 6 - A l r = A 1 t i - r + c o n s t a n t .i=1 i=1 i



D . C h h a j e d / E u r o p e a n J o u r n a l o f O p e r a t i o n a l R e s e a r c h 8 4 ( 1 9 9 5 ) 3 8 5 - 4 0 1 399

" ' _ Z ~ ( n 1, A 1, O) > Z ~ ( n 1, A 1 + 1 , O ) f o r A 1 > N - n v T h e r e f o r ei n c e n 1 s a t i s f i e s a l ) , E i = l t i < " r , and so

maxx lZ~( nl , A, 0) = m a x x l < N _ n l Z ' ~ ( n 1 , A 1, 0 ).

a 3 ) A g a i n b y L e m m a 2 , w h e n A 2 > 0 , i t is s u f f i c ie n t t o s e t ' ~ 1 = 0 a n d b y L e m m a 3 , i n t e g e r v a l u e s o f A 2

a r e s u f f i c ie n t . C o n s i d e r ( 1 3 ) a n d l e t

W i= - A 2 + i - 1 , i = 1 . . . . . n 1, t ) i = i - - 1 , i = l , . . . , N - n l .

N o t e th a t w , < w e < . " < w n ~ a n d v l < v 2 < " " < V N _ n . I f A 2 > n ~ - l , v x > w n . T h e r e f o re , t h e

l a r g e s t n , j o b s a r e a s s i g n e d t o t h e f i rs t d u e - d a t e . T h i s g i v es

nl N

Z ~ (( nl , O , A 2 ) = E W i tN - i + x + E v i - , l t N - i + l + ( 2 N - n i ) r 3 - A z ( ' r - T )i=1 i=n l+ l

( )A 1 T - - r - ~ t N _ i + 1 + c o n s t a n t .

i=1

A s E ' / ; lt N _ i + 1 > T - "r w h e n n I s a t i s f i e s a l ) , Z ~ (n 1, O, A 2) >_Z ~ ( n l , O, A 2 + 1 ) f o r A 2 > n 1 - 1 . T h e r e f o r e

m a x a Z * ( n l , 0, A z ) = m a x a : < n l _ l Z C ~ ( n l , O , } k 2 ) .

b l j T h e a r g u m e n t f o r t h e u p p e r b o u n d o n n ~ is s im i la r to th e c o r r e s p o n d i n g c a s e i n a l ) . T o s h o w th a tN - m 1 - 1 _< n ~ , n o t e t h a t t h e N - n I j o b s a s s i g n e d t o t h e s e c o n d d u e - d a t e m u s t b e s u c h t h a t t h e s u m o f

t h e i r p r o c e s s i n g t i m e s e x c e e d "r ( t o s a t is f y c o n s t r a i n t ( 1 6 ) ). m I i s t h e m a x i m u m n u m b e r o f j o b s w h i c h c a n

b e a s s i g n e d t o a d u e - d a t e w i t h o u t t h e i r t o t a l p r o c e s s i n g t i m e e x c e e d i n g "r a n d s o m I + 1 i s t h e m a x i m u m

n u m b e r o f j o b s w h i c h c a n b e a s s i gn e d t o t h e s e c o n d d u e - d a t e . T h u s , t h e m i n i m u m n u m b e r o f j o b s w h i c h

c a n b e a s s ig n e d to t h e fi rs t d u e - d a t e i s N - m ~ - 1 .

b 2 ) T h e p r o o f o f t h i s i s s i m i l a r t o a 2 ) . [ ]

A p p e n d i x C

W e f i r s t p r o v e a g e n e r a l r e s u l t w h i c h i s u s e f u l i n p r o v i n g T h e o r e m s 3 a n d 4 . L e t A = { a l , a 2 . . . a N}b e a s e t o f n o n - n e g a t i v e i n t e g e r s w i th a i >__a i÷ 1 , f o r a l l i = 1 , . . . , N - 1 , a n d B = { b 1 , b 2 . . . . b N} b e a s e t

o f n o n - n e g a t i v e r e a l n u m b e r s w i th b i <_ b i + 1 f o r a l l i = 1 . . . , N - 1 . G i v e n t w o s e t s a a n d /3 o f e q u a l

c a r d i n a l i t y , l e t C(o t , 3 ) d e n o t e t h e m i n i m u m v a l u e o f q ~ ( a ) "/3 w h e r e q ~ (a ) is a p e r m u t a t i o n o f a , a n d •

d e n o t e s t h e d o t p r o d u c t . L e t A ( i ) d e n o t e t h e s e t A w i t h a i r e p l a c e d b y a ~ + 1 . W e n o w h a v e t h e

f o l l o w i n g r e s u l t :

L e m m a C . 1 . I l k i i s the num b er o f e l em en t s o f A tha t a r e equa l to a i and h ave index s m a l le r than i i n A , i .e .,

k i = m a x { j : a i = a i _ f l , t hen C (A( i ) , B ) - C (A , B ) = b i - k , "

P r o o f . C ( A , B ) is o b t a i n e d b y m u l t i p ly i n g a i w i t h bi, f o r a l l i = 1 . . . , N , a n d a d d i n g t h e r e s u l t ( s e e

T a b l e C . 1 ) . W h e n a i i s r e p l a c e d b y a i + 1 , C ( A ( i ) , B ) is o b t a i n e d b y r e a r r a n g i n g A ( i ) i n n o n - i n c r e a s i n go r d e r a n d m u l t ip l y i n g t h e r e s u lt w i th B a s s h o w n i n T a b l e C . 1.

Table C.1

Rank 1 .. . i - k i - 1 i - k i i - k i + l . . . i - 1 i i+1 . . N

A a I " " " a i k i _ 1 a i k i a i _ k i + l • • " a i _ I a i a i + 1 • • a N

A ( i ) a ~ . . . a i _ k i _ 1 a i + l a i _ k , ' ' ' a i _ 2 a i _ 1 a i + 1 . . a N

B b l " ' " b i - k i 1 b i - k i b i - k i + l " ' " b i 1 b i b i + l "" b N



400 D. Chhajed European Journal of Operational Research 84 (1995) 385-401

I n t h e r e a r r a n g e m e n t o f A ( i ) , t h e p o s i t i o n s o f { a 1 . . . , a i _ k , _ l } a n d { a i + 1 . . . . aN} r e m a i n u n c h a n g e d .

E a c h o f { a i _ k i . . . . , a i - 1 } i s s h i f t e d t o t h e r i g h t b y o n e p o s i t i o n a n d a i i s s h i f t e d t o t h e l e f t b y k i p o s i t i o n s .

N o t e t h a t w h e n k i > 0 ,

i i i

Y'~ a j b j = ~ _, a j _ l b j = a i Y '. b j .

j=i-ki + l j=i-k i+ l j=i-ki + 1

T h u s , t h e o n l y d i ff e r e n c e b e t w e e n C ( A ( i ) , B ) a n d C ( A , B ) i s i n th e ( i - k i ) - t h t e r m . T h u s ,

C ( A ( i ) , B ) = C ( A , B ) - a i _ ~ b i _ ~ + ( a i + l ) b i _ ~ = C ( A , B ) + b i _ k . []

P r o o f o f T h e o r e m 3 . T h e L a g r a n g i a n d u a l f u n c t i o n ( 1 9) is a c o n c a v e f u n c t i o n o f A a n d s o a l o c al m a x i m a

i s g l o b a l m a x i m a . T h u s o n e w a y t o s o l v e ( 1 9 ) i s t o s o l v e ( 1 8 ) s t a r t i n g w i t h A = 0 a n d k e e p i n c r e a s i n g A b y

o n e ( o n ly i n t e g e r v a l u e s a r e s u f f ic i e n t b y L e m m a 3 ) u n t il t h e v a l u e o f (1 8 ) n o l o n g e r i n c r e a s es .

C o n s e q u e n t l y , t o p r o v e t h e t h e o r e m i t i s s u f f ic i e n t t o p r o v e t h a t Z ~ ( n l , A + 1) - Z ~ ' ( n l , ) t ) < 0 i f f A < ) t* .

F o r a g i v e n v a l u e o f A , d e f i n e C ( w , t ) = m i n ~_,iwit[i] wi t h w = A + i - 1 , i = 1 . . . n l , a nd w i = i - 1

f o r i = n~ + 1 . . . , N . Le t w ( I ) d e n o t e t h e s e t o f w e i g h t s w w i th a ll t h e w e i g h t s h a v i n g in d e x i n I

i n c r e a s e d b y 1 . F o r e x a m p l e , w (2 , 3 ) = { W l , w 2 + l , w 3 + 1 , w 4, . . . } . L e t { 1", 2 * . . . . , N * } b e t h e

p e r m u t a t i o n o f j o b s t h a t d e f i n e s t h e v a l u e o f C ( w , t ) .

W e n o w i n c r e a s e w nt b y o n e . N o t e t h a t w 1 < w 2 < w , , < w n, + 1 . A l s o , i f Wn, + 1 i s e q ua l t o w j ( j > n t ) ,

t h e r a n k o f Wn~ + 1 w i ll b e l o w e r t h a n t h e r a n k o f w j ( b y o u r c o n v e n t i o n o f a s s ig n i n g s m a l l e r j o b s t o t h e

f ir s t d u e - d a t e i n c a s e o f a ti e) . T h u s , k ,L = 0 a n d b y L e m m a C . 1, C ( w ( n l ) , t ) - C ( w , t ) = t t n , r w h e r e

[ n l ]* is t h e r a n k o f wn~ i n w w h e n a r r a n g e d i n n o n - i n c r e a s i n g o r d e r . A s n o t e d i n t h e p r o o f o f L e m m a

C . 1 , i n c r e a s i n g Wn~ d o e s n o t c h a n g e t h e jo b s a s s i g n e d t o a n y o t h e r w i < wnl.

W e n e x t i n c r e a s e W n l _ 1 t o W n l _ 1 + 1 a nd t h i s w i l l g i ve C(w( n I - 1,n~), t ) - C ( w ( n l ) , t ) = t l, l _ q .

C o n t i n u i n g t h i s w a y b y i n c r e a s i n g w , ~_ 2 , w n ~_3 . . . . w ~ w e g e t ,

C ( w ( 1 , . . . , n , ) , t ) - C ( w ( 2 , . . . , n , ) , t ) = t i l l . .

T h e r e f o r e ,

C ( w ( 1 . . . . . n , ) , t ) - C ( w , t ) = t [ ,] . + " - " + t [ , , ] . = t ( F ) .

N o t e t h a t C ( w ( 1 . . . . , n t ) , t ) is e q u a l t o t h e f ir s t t w o t e r m s o f Z ~ ( n 1 , A + 1 ) . T h i s g i v e s

Z ~ ( n , , A + 1) - Z t ( n t , A ) = t ( F ) - ( T - ' r ) .

T h u s , i t i s b e t t e r t o i n c r e a s e X t o , t + 1 i f f t ( F ) e x c e e d s T - ~ '. [ ]

R e f e r e n c e s

A h m a d i , R . , a n d B a g c h i , U . (1 9 86 ), " S i n g l e m a c h i n e s c h e d u l i n g to m i n i m i z e e a r li n e s s s u b j e c t t o d e a d l i n e s " , W o r k i n g P a p e r8 6 / 8 6 - 4 - 1 7 , D e p a r t m e n t o f M a n a g e m e n t , U n i v e r s it y o f T e xa s , A u s t in , T X .

B a g c h i , U . , S u l l iv a n , R .S . , a n d C h a n g , Y .L . ( 19 8 6) , " M i n i m i z i n g m e a n a b s o l u t e d e v i a t i o n o f c o m p l e t i o n t i m e s a b o u t a c o m m o n d u e

d a t e " , Naval Research Logistics Quarterly 33 , 227-240 .

B a k e r , K .R . , a n d S c u d d e r , G .D . ( 1 99 0 ), " S e q u e n c i n g w i t h e a r l i n e s s a n d t a r d i n e s s p e n a l t i e s : A r e v i e w " , Operations Research 38 ,

2 2 - 3 6 .C h a n d , S . , a n d C h h a j e d , D . ( 19 92 ), " D e t e r m i n a t i o n o f o p t i m a l d u e d a t e s a n d s e q u e n c e " , Operations Research 4 0 , 5 9 6 - 6 0 2 .

C h a n d , S . , a n d S c h n e e b e r g e r , H . ( 1 9 8 4 ) , " S i n g l e m a c h i n e s c h e d u l i n g t o m i n i m i z e w e i g h t e d e a r l i n e s s s u b j e c t t o n o t a r d y j o b s " .

European Journal of Operational Research 34 , 221-230 .F i s h e r , M .L . ( 1 98 1 ), " T h e l a g r a n g i a n r e l a x a t i o n m e t h o d f o r s ol v i n g i n t e g e r p r o g r a m m i n g p r o b l e m s " , Management Science, 27 ,

1 - 1 8 .



D. Chhajed European Journal of Operational Research 84 (1995) 385-401 40 1

G a r e y , M .R . , T a r j a n , R .E . a n d W i l f o n g , G .T . ( 1 98 8 ), " O n e - p r o c e s s o r s c h e d u l i n g w i t h e a r l i n e s s a n d t a r d i n e s s p e n a l t i e s " ,

Mathematics o f Operations Research 1 3 , 3 3 0 - 3 4 8 .

H a l l , N .G . , K u b i a k , W . , a n d S e t h i , S.P. ( 19 9 1) , " E a r l i n e s s - t a r d i n e s s s c h e d u l i n g p r o b l e m s , I I: D e v i a t i o n o f c o m p l e t i o n t i m e s a b o u t a

r e s t r i c t i v e c o m m o n d u e d a t e " , Operations Research 3 9 , 8 4 7 - 8 5 6 .

K a n e t , J . J . ( 19 8 1) , " M i n i m i z i n g t h e a v e r a g e d e v i a t i o n o f j o b c o m p l e t i o n t i m e s a b o u t a c o m m o n d u e d a t e " , Naval Research Logistics

Quarterly 28 , 643-651 .

M a t s u o , H . I . ( 1 9 8 8 ) , " T h e w e i g h t e d t o t a l t a r d i n e s s p r o b l e m w i t h f i x e d s h i p p i n g t i m e s a n d o v e r t i m e u t i l i z a t i o n " , Operations

Research 3 6 , 2 9 3 - 3 0 7 .

N e m h a u s e r , G .L . a n d W o l s e y , L .A . ( 1 9 8 8 ) , Integer and Combinatorial Optimization, W i l e y , N e w Y o r k .

P a n w a l k e r , S .S ., S m i t h , M .L . a n d S e i d m a n n , A . (1 9 82 ), " C o m m o n d u e d a t e a s s i g n m e n t to m i n i m i z e t o t a l p e n a l t y f o r t h e o n e

m a c h i n e s c h e d u l i n g p r o b l e m " , Operations Research 30 , 391-399 .

fixed interval due date

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