Job Sequencing Rules for Minimizing Makespam

8/2/2019 Job Sequencing Rules for Minimizing Makespam

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EUROPEANJOURNALOF O PERATIONALRESEARCHELSEVIER Europe a n J ourna l o f O pe ra t iona l R e s e a rc h 96 (1996) 274 -288

Theory and MethodologyJob sequencing rules for m inimizing the expected m akespan in

f lexib le machinesAmiya K. Chakravarty a , Nagra j Balakris lman u, ,

a A.B. Freeman Sctu~ol ofBuMness, Tulane University, New Orleans, LA 70118, USAb Department of Management, Clemson Unioersity, Clemson, SC 29634, USA

R e c e iv e d 1 Aug us t 1994; a c c e pte d 1 O c tobe r 1995

A b s t r a c tW e consider scheduling o f a deteriorating flexible m achine that is capa ble of processing a num ber of diverse jobs w ithnegligible setup times between job s. Specifically, w e deve lop rules for sequencing N jo bs on such a machine such that itsexpected ma kesp an (sum of all job processing tim es and machine down-tim e) is minimized. U sing the Weibull distributionto characterize machine failures in our model, we permit different jobs to contribute to machine deterioration (and hence itsfailure) at different failure rates, and do not require these rates to remain constant with machine-use time. We validate theeffectiveness of these job sequencing rules for different cases, using extensive simulation tests .

Keywords: M a c hine f a ilu re ; M a c h ine r e l i ab i l i ty ; J ob s c he d ul ing ; Ex pe c te d m a k e s pa n; F le x ib le ma c hine s

1 . In t r o d u c t i o nI n j o b s c h e d u l i n g p r o b l e m s w h e r e e a c h j o b i s p r o c e s s e d o n s e v e r a l m a c h i n e s , m i n i m i z a t i o n o f m a k e s p a n

( d e f in e d a s t h e s u m o f jo b p r o c e s s i n g a n d s e t u p t i m e s ) a l s o m a x i m i z e s m a c h i n e - u t il i z at i o n . M i n i m i z a t i o n o fm a k e s p a n , h o w e v e r , i s t y p i c a l l y n o t a n o b j e c t i v e i n s i n g l 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 s , a s n o w m a k e s p a n i si n d e p e n d e n t o f j o b s e q u e n c e ( a s s u m i n g s e q u e n c e - in d e p e n d e n t s e t u p t i m e s ).

I n t h e c a s e o f a f l e xi b l e m a c h i n e c a p a b l e o f p r o c e s s i n g a n u m b e r o f d i v e r s e j o b s w i t h n e g l i g i b le s e t u p t i m e sb e t w e e n j o b s , m a k e s p a n i s j u s t t h e s u m o f a l l j o b p r o c e s s in g t i m e s . H o w e v e r , i f m a c h i n e s t o p p a g e s o c c u r b e fo r et h e c o m p l e t i o n o f a l l j o b s , m a k e s p a n w o u l d o b v i o u s l y b e e x t e n d e d b y t h e t o t a l m a c h i n e d o w n - t i m e . T h ep r i m a r y c a u s e s o f s u c h d o w n - t i m e a r e t o o l - m a g a z i n e c h a n g e s a n d m a c h i n e f a i lu r e s . W h i l e t h e i m p a c t o f j o bs e q u e n c e o n t o o l - m a g a z i n e c h a n g e s h a s b e e n w e l l d o c u m e n t e d [ 5 , 2 0 - 2 3 ] , t h e r el a t io n s h i p b e t w e e n j o b s e q u e n cea n d m a c h i n e f a i l u r e s h a s n o t b e e n a d d r e s s e d .

I n p r a c t i c e , t h e m i n i m i z a t i o n o f m a k e s p a n i n a f l e x i b l e m a c h i n e , w h i c h i s e q u i v a l e n t t o m i n i m i z a t i o n o fm a c h i n e d o w n - t i m e d u e t o u n e x p e c t e d f a i l u r e s , w o u l d b e a n i m p o r t a n t o b j e c t i v e f o r s e v e r a l r e a s o n s . R e c e n ts u r v e y s [ 1 8 ,2 4 ] h a v e s h o w n t h a t c o n t r o l l i n g u n p l a n n e d m a c h i n e d o w n - t i m e d u e t o f a i l u r e s i s e s p e c i a l l y c r i ti c a lf o r f l e xi b l e m a c h i n e s w h i c h i n c u r f o u r t i m e s t h e w e a r o f d e d i c a te d m a c h i n e s d u r i n g a g i v e n t i m e i n te r v a l, d u e

* Corresponding author . Fax: 864-656-2015.0 3 7 7-2 217 /96 /$1 5 .0 0 C opy r ig h t 1996 E l s e v ie r Sc ie nc e B .V. Al l r ig h t s r e se rv e d .PH S 0 3 7 7 - 2 2 1 7 ( 9 6 ) 0 0 0 7 0 - 7


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A.K. Chakravarty , N . Balakrishnan / European Journa l o f Operational Research 96 (1996) 27 4-2 88 275

t o i n c r e a s e d l o a d . T h e e f f e c t o f s u c h a c c e l e r a t e d w e a r c l e a r l y e n h a n c e s t h e i m p o r t a n c e o f p r e v e n t i v em a i n t e n a n c e f o r s u c h m a c h i n e s [ 1 3] , a n d t h e r e h a v e b e e n m a n y s t u d ie s o n t h e o p t im a l t i m i n g f o r m a i n t e n a n c e o fde t e r io ra t ing m ach ines [ 15 ,17]. K enn edy [10] s t a t e s t ha t the ind i rec t cos t o f unplanned m a c h i n e d o w n - t i m e i sm u c h g r e a t e r t h a n t h a t f o r s c h e d u l e d m a i n t e n a n c e , a n d i s t h e m o s t s i g n i f i c a n t c o s t o f t h e s y s t e m . F u r t h e r , af l e x i b l e m a c h i n e t y p i c a l l y h a s a h i g h p r e m i u m o n p r o d u c t i v i t y d u e t o th e c o m p l e x i t y o f th e m a c h i n e w h i c h n o wi n c l u d e s n o t o n l y m e c h a n i c a l p a r ts b u t a l s o e l e c t r o n i c, h y d r a u l i c , s o f t w a r e , a n d h u m a n e l e m e n t s [ 2 4 ] - a n i s s u et h a t c l e a r l y a r g u e s f o r m i n i m i z a t i o n o f m a c h i n e d o w n - t i m e d u e t o f a i l u r e s . F i n a l l y , i f t h e f l e x i b l e m a c h i n ep r o d u c e s a l l c o m p o n e n t s f o r a n a s s e m b l y , t h e s e c o m p o n e n t s m u s t b e r e a d y b y t h e s c h e d u l e d a s s e m b l y s t a r tt i m e , w h i c h w o u l d a g a i n c a l l f o r m i n i m i z a t i o n o f c o m p o n e n t m a k e s p a n , e s p e c i a l l y i n v e r t i c a l l y i n t e g r a t e d J I Tsys t ems [8 ] .

S i n c e a f l e x i b l e m a c h i n e p e r f o r m s d i f f e r e n t m a n u f a c t u r i n g o p e r a t i o n s f r o m j o b t o j o b , t h e m a c h i n ed e t e r i o r a t i o n d u e t o e a c h j o b w o u l d b e a f u n c t i o n o f t h a t j o b ' s e n g i n e e r i n g - d e t e r m i n e d p a r a m e t e r s s u c h a s t h ej o b f e e d r a t e , d r il l in g o r c u t t i n g s p e e d , c h i p r e m o v a l r a t e , c o o l i n g e f f i c i e n c y f o r t h e t y p e o f c u t , e t c . E s s e n t i a l ly ,e a c h j o b , b a s e d o n t h e c o m p l e x i t y a n d s p e e d o f i t s o p e r a t i o n s , m a y b e v i e w e d a s p l a c i n g a c e r t a i n a m o u n t o f" s t r e s s " o n t h e m a c h i n e , a n d t h e m a c h i n e f a i l s w h e n t h e to t a l s t re s s e x c e e d s s o m e t h r e s h o l d v a l u e . S u c h s h o c kfa i lu re mode l s have been used extens ive ly in t he l i t e ra tu re [3] . T he s t re ss accen tua te s t he fa i lu re ra t e d i f fe ren t i a l si n t h e c o m p o n e n t s o f t h e m a c h i n e [ 2 4 ] a n d d i f f e r e n t j o b s w o u l d , t h e r e f o r e , c o n tr i b u t e to m a c h i n e d e t e r i o r a t io n( a n d h e n c e , i t s f a i l u r e ) a t d i f f e r e n t r a t e s . O b s e r v e t h a t t h e f a i l u r e r a t e d u e t o e a c h j o b m a y e i t h e r b e c o n s t a n tove r t ime o r i nc reas ing wi th mach ine -use t ime [2 ] . In e i the r s i t ua t ion , a s we wi l l e s t ab l i sh l a t e r , t he f requency o fm a c h i n e f a i l u r e s ( a n d h e n c e , m a c h i n e d o w n - t i m e s f o r r e p a i r s ) w o u l d b e j o b s e q u e n c e d e p e n d e n t .I n t h i s p a p er , w e s t u d y t h e r e la t i o n s h i p b e t w e e n j o b s e q u e n c e a n d m a k e s p a n a s s u m i n g t h a t t h e t o o l s re q u i r e df o r p r o c e s s i n g a l l j o b s f i t i n a s i n g l e t o o l - m a g a z i n e ( i . e . , t h e r e i s n o m a c h i n e d o w n - t i m e f o r c h a n g i n g t h et o o l - m a g a z i n e ) . S p e c i f i c a ll y , w e d e v e l o p r u l e s f o r s e q u e n c i n g N j o b s o n a d e t e r i o r a t in g f l e x i b le m a c h i n e s u c ht h a t it s e x p e c t e d m a k e s p a n i s m i n i m i z e d . T h e m a c h i n e f a i l u r e d i s t ri b u t io n a s s o c i a t e d w i t h a j o b i s a s s u m e d t o b ewe ibu l l , so t ha t bo th cons t an t and inc reas ing fa i lu re ra t e scena r ios can be ana lyzed . In a fo l low-up pape r [4 ] ,u s i n g t h e r e s u l t s o f t h i s r e s e a r c h , w e s t u d y t h e i m p a c t o f t h e j o b s e q u e n c e o n b o t h e x p e c t e d m a k e s p a n a n dn u m b e r o f t o o l- m a g a z i n e c h a n g e s.A n u m b e r o f a u t h o r s h a v e a n a l y z e d d e t e r i o r a t in g n o n - f l e x i b l e p r o d u c t i o n s y s t e m s [ 1 2 ,1 5 , 1 9 ]. B e c a u s e o f i tsana ly t i ca l s impl i c i t y , t he ma ch ine fa i lu re d i s t r i bu t ion i s cha rac t e r i zed by the expon en t i a l d i s t r i bu t ion [6 , 11 , 16 , 17] .W hi l e a f ew au thors [ 1 ,16] have a t t em pted to jus t i fy use o f t he exponen t i a l d i s t r ibu t ion , o the r s [2 , 7 , 14] po in t ou tt h a t t h e r e s u l t s o b t a i n e d a r e l i k e l y t o b e i n v a l i d a n d o v e r l y o p t i m i s t i c , e s p e c i a l l y i f t h e r e a r e c o m p o n e n t s t h a tcon t r ibu te t o mach ine fa i lu re a t i nc reas ing ra t e s . A typ ica l ob jec t i ve in such s tud ie s i s t o f i nd the op t ima l t imingf o r i n s p e c t i o n a n d m a i n t e n a n c e o f d e t e r i o r a t i n g s y s t e m s s u c h t h a t t h e a v e r a g e c o s t o f t h e r e p a i r p o l i c y i sm i n i m i z e d i n t h e l o n g r u n . T h e e f f e c t o f m u l t ip l e jo b s , c o n t r i b u ti n g t o m a c h i n e d e t e r i o r a t i o n a t d i ff e r e n t r at e s ,h a s n o t b e e n a d d r e s s e d . F u r t h e r , c a s e s w h e r e t h e s e r a t e s i n c r e a s e w i t h t i m e h a v e a l s o n o t b e e n c o n s i d e r e d .T h i s p a p e r i s o r g a n i z e d a s f o l l o w s . I n S e c t i o n 2 , w e d e s c r i b e t h e t h e o r e ti c a l b a s i s o f o u r a n a l y s e s . I n S e c t i o n3 , w e f i r s t d e v e l o p t h e o p t i m a l j o b s e q u e n c i n g r u l e a ss u m i n g t h e m a c h i n e f a i l u re r a te d u e t o e a c h j o b i s c o n s t a n to v e r t i m e , a n d t h e n u s e s i m u l a t i o n t e s t s t o s h o w i t s e f f e c t i v e n e s s . I n S e c t i o n 4 , w e d e v e l o p t h e o p t i m a l j o bs e q u e n c i n g r u l e f o r p r o b l e m s w h e r e t h e s e m a c h i n e f a i l u r e r a t e s i n c r e a s e w i t h m a c h i n e - u s e t i m e , a n d a g a i n u s es i m u l a t i o n t es t s t o s h o w i ts e f f e c t i v e n e s s . I n S e c t i o n 5 , w e p r e s e n t o u r c o n c l u s i o n a n d d i s c u s s e x t e n s i o n s t o t h i sp r o b l e m .

2 . E x pec t ed m a kes pa n a nd i t s equ i v a l en t sI f t h e re a r e n m a c h i n e f a i l u r e s b e f o r e a f l e x i b l e m a c h i n e p r o c e s s e s a l l j o b s , a n d t h e d o w n - t i m e f o r r e p a i r s a t

e a c h f a i lu r e is a s s u m e t o b e R , t h e m a k e s p a n ( M S ) o f t h e j o b - m i x i s g iv e n b y :M S = T+ nR, ( 1 )


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276 A.K. Chakravarty, N. Balakrishnan / European Journal o f Operational Research 96 (1996) 274-288w h e r e T i s th e s u m o f a l l j o b p r o c e s s i n g t i m e s . N o w , i f w e l e t p (n ) d e n o t e t h e p r o b a b i l i t y o f n f a i l u r e s b e f o r et h e c o m p l e t i o n o f a l l j o b s , t h e e x p e c t e d m a k e s p a n ( E M S ) c a n b e w r i t t e n a s :

E M S = r ,~ = 0 [ r + nR] p ( n ) = r + R Z ; . , n p ( n ) . ( 2 )O b s e r v e t h a t p ( 0 ) i s t h e p r o b a b i l i t y t h a t t h e m a k e s p a n e q u a l s T . W e c a l l t h i s m e a s u r e t h e m a k e s p a n

p r o b a b i l i t y , M S P ( i . e . , p ( 0 ) = M S P ) . A s w e w i l l s h o w l a t e r , t h e m a t h e m a t i c a l e x p r e s s i o n f o r M S P , a l t h o u g h i tr e q u i r e s r a t h e r c o m p l e x r e a s o n i n g t o e s t a b l i s h , i s t r a c t a b l e . T h e m a t h e m a t i c a l e x p r e s s i o n s f o r p(n) , h o w e v e r ,b e c o m e i n c r e a s i n g l y i n t r a c ta b l e a s t h e v a l u e o f n i n c r ea s e s . T h e r e f o r e , i n o r d e r t o f a c il i ta t e f u r th e r m a t h e m a t i c a la n a l y s i s o f j o b s e q u e n c i n g r u l e s , w e e x p l o r e t h e s u b s t it u t io n o f t h e o b j e c t i v e fu n c t i o n ( m i n i m i z a t i o n o f E M S ) b ya s u r r o g a t e f u n c t i o n t h a t w o u l d : ( i ) b e m o n o t o n i c a l l y r e l a te d t o E M S ; a n d ( i i ) b e m a t h e m a t i c a l l y tr a c ta b l e . S i n c ea s t r o n g i n v e r s e r e l a t i o n s h i p w o u l d e x i s t b e t w e e n E M S a n d M S P , m a x i m i z a t i o n o f M S P c a n b e e s t a b l i s h e d a sb e i n g e q u i v a l e n t t o m i n i m i z a t i o n o f E M S . O b s e r v e t h a t m a x i m i z a t i o n o f M S P w o u l d a l s o b e e q u i v a l e n t t om a x i m i z a t i o n o f t h e p r o b a b i li t y th a t m a k e s p a n i s w i t h i n s o m e t a r g e t ti m e w i n d o w s u c h a s T a n d T + R . S u c h a no b j e c t i v e m a y b e r e l e v a n t , f o r e x a m p l e , i f s o m e c o m m o n d u e d a t e e x i s t s f o r t h e c o m p l e t i o n o f a l l j o b s .

I n w h a t f o l l o w s , w e e s t a b l i s h a m a t h e m a t i c a l e x p r e s s i o n f o r M S P a s s u m i n g m a c h i n e f a i l u r e s t o b ec h a r a c t e r i z e d b y t h e w e i b u l l d i s tr i b u ti o n . T h e w e i b u l l d i s tr i b u t io n p r o v i d e s a m o r e g e n e r a l i z e d a p p r o a c h i n t h ea n a l y s i s o f m e c h a n i c a l s y s t e m s [ 2 ] t h a n t h e e x p o n e n t i a l d i s t r i b u t i o n s i n c e t h e w e i b u l l P D F c a n b e s h a p e d t or e p r e s e n t d i f f e r e n t d i s t r i b u t i o n s b y a d j u s t i n g i t s p a r a m e t e r v a l u e s . I t m a y b e o f i n t e r e s t t o n o t e t h a t a m o d e lb a s e d o n t h e w e i b u l l d i s t r ib u t i o n i s u s e d t o e s t i m a t e c o m p u t e r s o f t w a r e f a i l u r e s [9 ]. T h e W e i b u l l f a i l u r e d e n s i t yfunc t ion i s g iven a s [2 ] :

f ( t ) = ( f i t tJ- ' /o r ~) e x p ( - - t / a ) t~ ( 3 )whe re t i s t he e l apsed t ime , a i s t he sca l e pa ram e te r , and 13 i s the shape pa ram e te r . I t i s we l l kno wn [1] t ha tthe re i s a s i gn i f i can t d i f fe rence in t he shape o f t he PDF curv e fo r 13 = 1 (which i s t he exponen t i a l d i s t r ibu t ion)and 13 > 1 . So m e of t he re su l t s we de r iv e in l a t e r sec t ions a re a l so m arke d ly d i f fe ren t fo r t hese two s i tua t ions . I thas a l so be en e s t ab l i shed [2 ] t ha t 13 < 1 dur ing the " w ea r - in " phase o f a mach ine , 13 = 1 dur ing the s t eady o r" n o r m a l l i f e " p h a s e , a n d 13 > 1 w h e n " w e a r - o u t " p h a s e s e t s i n . M o r e o v e r , f o r m e c h a n i c a l c o m p o n e n t s , t h enorm a l l i f e (13 = 1 ) phase i s ex t rem e ly na r row [2].

T o p e r m i t d i f f e r e n t j o b s t o c a u s e m a c h i n e d e t e r i o ra t i o n a t d i f f e r e n t r a te s , t h e m a c h i n e f a i l u r e d i s tr i b u ti o n d u et o j o b i m a y b e r e p r e s e n t e d u s i n g a j o b - d e p e n d e n t s c a le p a r a m e t e r c ( ; [ 2] . S i n c e 13 o n l y d e s c r i b e s t h e s h a p e o ft h e P D F ( b a s e d o n t h e m a c h i n e ' s o p e r a t i n g p h a s e ) , i t w o u l d p o s s e s s a f i x e d v a l u e f o r a g i v e n p r o b l e m s c e n a r i o ,i n d e p e n d e n t o f j o b s . O b s e r v e t h a t a t a t r a n s it i o n f r o m o n e j o b t o a n o t h e r , t h e f a il u r e d is t r ib u t i o n s w i t c h e s f r o mo n e P D F t o a n o t h e r . H o w e v e r , e a c h s u c c e e d i n g P D F m u s t b e c o n d i t i o n e d u p o n t h e m a c h i n e s u r v i v i n g u p t o t h a tp o i n t in t i m e , h a v i n g p r o c e s s e d s o m e m i x o f j o b s .

T o u n d e r s t a n d t h e e f f e c t o f j o b s w i t c h e s o n t h e m a c h i n e , c o n s i d e r F i g . 1 w h i c h s h o w s t h e W e i b u l l P D F f o rjobs i and j (wi th o t i < o r , and 13 = 3) . I f job i p rec ede s job j , t he fa i lu re d i s t r i bu t ion be tw een t im e 0 and t i isg o v e r n e d b y f ~ ( t ) a s s h o w n b y t h e l ig h t l y - s h a d e d r e g i o n . A t t i m e t i, t he m a c h i n e s w i t c h e s f r o m j o b i t o j o b j .D u e t o t h e m a c h i n e ' s f l e x i b l e n at u r e , th i s s w i t ch i s i n s t a n t a n e o u s a n d n o a d j u s t m e n t s a r e m a d e t o t h e m a c h i n e .N o w , f r o m t i m e t i t o t i + tj, the m a c h i n e f a i l u r e d i s t r i b u t i o n i s g o v e r n e d b y f f l t ) ( s h o w n b y t h e d a r k l y - s h a d e dreg ion) , cond i t i oned on the even t t ha t t he mach ine d id no t f a i l t i l l t ime t~ .

O b s e r v e t h a t t h e c o n d i t i o n i n g o f t h e f a i l u r e d i s t r i b u t i o n f f l t ) s h o u l d r e f l e c t t h e f a c t t h a t t h e m a c h i n ede te r io ra t ion be tw een t im e 0 and t~ i s due to job i , and no t job j . T ha t i s , a s t he m ach in e ins t an taneou s lys w i tc h e s f r o m o n e j o b t o a n o th e r , th e r e is a c o n t i n u o u s " c a r r y - f o r w a r d " o f m e m o r y w h i c h w o u l d e x i s t e v e nwhen the mach ine fa i lu re d i s t r i bu t ions wi th ind iv idua l jobs a re exponen t i a l ( i . e . , t he p rocess i s no t t o t a l l ym e m o r y - l e s s ) .I n A p p e n d i x A , w e d e v e l o p th e e x p r e s si o n s h o w n b e l o w f o r P ( i , j ) , d e f i n e d a s t h e p r o b a b i l it y o f at least onemachine fai lure b e f o r e t h e c o m p l e t i o n o f b o t h j o b s i a n d j , p r o c e s s e d i n t h at o r d e r:

p ( i , j ) = ( f o , f . ( t ) d t W f t [ , + t , f j ( t ) d t ) / ( f o t , f ~ ( t ) d t + J.ft'ti+ j f j ( t ) dtW ft,+ tfD (t)dt),~ ( 4 )


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A.K. Chakravarty, N. Balakrishnan / European Journal of Operational Research 96 (1996) 274-288 2 7 7

AIItII4-}~4

0P4

t ,

f i ( t )

I I I I

t i + t j t i m e . . . . >F i g . 1 . W e i b u l l P D F f o r [3 = 3 a n d d i f f e r e n t t x v a l u e s .

j ( t )

w h e r e f D ( t ) i s t h e m a c h i n e f a i l u r e d is t ri b u ti o n a s s o c i a te d w i t h a d u m m y j o b a t th e e n d o f a n y j o b s e q u e n c e ( s e eA p p e n d i x A f o r a n e x p l a n a t i o n ) . F r o m t h e a b o v e e x p r e s s i o n , w e n o t e t h a t f o r a ll [3 v a l u e s , P ( i , j ) d e p e n d s o nt h e o r d e r i n w h i c h t h e j o b s a r e p r o c e s s e d .L e t P M ( i , j) d e n o t e t h e M S P o f p r o c e s s i n g j o b s i a n d j i n t h a t o r d e r . S u b s t i t u t in g f o r f / ( t ) , ~ ( t ) , a n d f D ( t ) inE q . ( 4 ) , w e o b t a i n :

i j ) = l - P ( i , j ) = e x p ( - [ t, +

/ ( l - e x p ( t ,/ a ,) " e x p f - t , / a , ) " - e x p ( - [ t , t j l/ a :) " e x p ( - [ t , t j l / a o ) ' .( 5 )

A s i m i la r a n a l y s i s m a y b e u s e d t o c a lc u l a t e th e M S P o f p r o c e s s i n g a n y s e q u e n c e i n v o l v i n g m u l t ip l e j o b s . L e tt d e n o t e t h e p r o c e s s in g t i m e o f j o b i . I f j o b s 1 t o N a r e p r o c e s s e d i n th e s e q u e n c e 1 ,2 . . . . . N , a n d Tk = ~ = t t i,t h e n t h e M S P b e e x p r e s s e d a s :

P M ( 1 ,2 . . . . . N ) = e x p ( - -T N / O t D ) ~/ ( 1 - - . Y f f = , e x p ( - - T k / a k ) ~ + ~ f f _ _ - l ' e x p ( - - T k / a k + , ) / ~ + e x p ( - - T N / a O ) ~ ) . ( 6 )

I n w h a t f o l l o w s , f o r g i v e n v a l u e s o f t h e p a r a m e t e r s [ 3, u i , a n d t ~, w e d e v e l o p j o b s e q u e n c i n g r u l e s tom a x i m i z e t h e M S P .


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2 7 8 A.K. Chakravarty, N. Balakrishnan / European Journal o f Operational Research 96 (1996) 274-2883 . S e q u e n c i n g r u l e s w i t h c o n s t a n t f a i l u r e r a t es

I n th i s s e c ti o n , w e i d e n t i f y t h e j o b s e q u e n c e t h a t m a x i m i z e s M S P , a s s u m i n g t h e m a c h i n e f a i l u r e r a t e d u e t oeach j ob i s cons tan t o ve r t ime ( i . e . , 13 = 1 ) . A l l j obs a re a s s um ed to be ava i l ab le a t t ime t = 0 . W e f i r s t cons ide ra n i n t e r c h a n g e o f j o b s i n a 2 - j o b s e q u e n c e . T h e f o l l o w i n g t h e o r e m n o w a p p l ie s :T h e o r e m 1 . I f 13 = 1 , a nd job s j and k are d e f ined such tha t c ry < e t k , t hen P m( k , j ) >_ P M( j , k ) , i . e. , t he j ob ssh o u l d b e p ro c e s se d i n n o n - i n c r e a s i n g o rd e r o f c t .P r o o f . A s s u m e P m ( k , j ) < P M ( j ,k ) . E x p r e s s i n g P m ( j , k ) an d P M ( k , j ) us ing E q . (5 ) , i t c an be s how n tha t thef o l l o w i n g i n e q u a l i t y m u s t h o l d f o r t h i s a s s u m p t i o n t o b e t r u e :

e x p ( - t , / a 2 ) - e x p( - t k / a k ) - e x p ( - [ t j + t , ] / a j ) + e x p ( - t j / % )- e x p ( - t j / a , ) + e x p ( - [t~ + t , l / a , ) > O . ( 7 )

L e t e x p ( - t j / c t j ) = a a n d e x p ( - t k / e t k ) = b . Sin ce ot j < a k , exp( - tJ o t k) and ex p( - t k / e t i ) m a y b e w r i t t e n a s( a + h ) an d ( b - m ) r e s p e c t i v e l y , w h e r e h , m > O . E q . ( 7 ) m a y t h e n b e w r i t t e n ( a f t e r s i m p l i f i c a t i o n ) a s :

h ( b - 1 ) + r e ( a - 1 ) > 0 , ( 8 )w hich canno t be t rue s ince 0 < a , b < 1 . H ence , P M ( k , j ) > P M ( j , k ) . [ ]

U s ing T h eo rem 1 , i t is pos s ib le to op t im a l ly s eq ue nce pa i r s o f j obs . W e no te tha t w hen 13 = 1 , e t 2 den o te s them e a n t i m e b e t w e e n f a i l u r e s f o r t h e m a c h i n e f a i l u r e d i s tr i b u t i o n d u e t o j o b j , a n d w o u l d d e p e n d o n t h e s tr e s s o fj o b j o n t h e m a c h i n e . H e n c e , a c o n c ep t u a l i n t er p r e ta t io n o f T h e o r e m 1 m a y b e t o p r o c es s t he " s i m p l e r " j o bf i rs t . N e x t , w e e x a m i n e w h e t h e r t h e a b o v e r u l e h o l d s e v e n w h e n t h e r e ar e m o r e t h a n t w o j o b s i n t h e s e q u e n c e .T h e o r e m 2 . I f 13 = 1 , and j obs j and k are such tha t c t i < ct k , t hen i t i s op t imal t o process j ob k be fore job j ,p ro v i d e d t h e to t a l p ro c e s s i n g t i m e o f a l l j o b s p re c e d i n g t h is p a i r d o e s n o t e x c e e d a n o n - n e g a t i ve t h re sh o l d T B ,where :

T 8 = 4 , - log ( ( [ 1 - ex p ( - t J a , )] [ 1 - e x p ( - t , / 4 , ) 1 ) / ( [ 1 - e x p ( - t , / 4 , ) l [ 1 - e x p ( - t j / a , ) 1 ) } . (9 )

P r o o f . C o n s i d e r s e q u e n c e s ( 1 . . . . . i , ( j , k ) , m . . . . . N ) a n d ( 1 . . . . . i , ( k , j ) , m . . . . . N ) . L e t T / b e t h e su m o fp r o c e s s i n g t i m e s o f a l l j o b s f r o m j o b 1 t o j o b i , i .e . , a ll j o b s p r e c e d i n g t h e j o b p a i r ( j , k ) .

A s s u m e P M ( 1 . . . . . i , (k , j ) . . . . . N ) < P M ( 1 . . . . . i , ( j , k ) . . . . . N ) , w h e r e T i < T B . E x p r e s s i n gP M (1 . . . . . i , (k , j ) . . . . . N ) a nd P M (1 . . . . . i , ( j , k ) . . . . . N ) u s i n g E q . ( 6 ) , i t c a n b e s h o w n t h a t t h e f o l l o w i n gi n e q u a l i t y m u s t h o l d f o r t h i s a s s u m p t i o n t o b e t r u e :

e x p ( - T ~ / a k ) - e x p ( - - [T ~ + t , ] / 4 , ) + e x p ( - - [ T , + t , ] / % ) - e x p ( - [ T ~ + t k + t j ] / a 2 )> e x p ( - T ~ / % ) - e x p ( - [ T ~ + t j ] / % ) + e x p ( - [ T i + t j ] / a k ) - e x p ( - [ T ~ + t j + t ~ ] / 4 ~ ) ( 1 0 )

w h i c h , a f t e r r e a r r a n g i n g t h e t e r m s , b e c o m e s :e x p ( r / [ , , , -

( ( [ 1 - e x p ( - t k / 4 , ) ] [ 1 - e x p ( - t J a k ) ] ) / ( [ 1 - e x p ( - t J a j ) ] [ 1 - e x p ( - t k / a 2 ) ] ) ) > 1 . (11 )T his ca n be s im pl i f i ed to 7 ], > T B , w hich i s a con t rad ic t ion . H ence , i f the to ta l p roce s s ing t im e T o f a ll j obs

p r e c e d i n g t h e p a i r ( j , k ) i s < T B , i t i s op t im a l to p roce s s j ob k be fo re j ob j . [ ]


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A.K . Chakravarty, N. Balakrishnan / European Journal of Operational Research 96 (1996) 27 4-2 88 279R e m a r k 1 . W e n o t e t h a t si n c e o tj < o r , , t h e t h r e s h o l d T B i s n o n - n e g a t i v e . F u r th e r , f o r e a c h j o b p a i r ( j , k ) , T B i sa f un c t i on o f t j , t , , o r , and o t k .3 . 1. L o w e r b o u n d o n t h r e s h o ld T B

I n w h a t f o l l o w s , w e i n v e s ti g a t e t h e l o w e r l i m i t o n T B i n E q . (9 ) . O u r o b j e c t i v e i s to d e t e r m i n e a n u p p e rb o u n d o n T ~ t h a t w o u l d e n s u r e v a l i d i t y o f th e s e q u e n c i n g r u l e e s ta b l i sh e d i n T h e o r e m 2 , a s lo n g a s a p r o d u c t i o nr u n d o e s n o t e x c e e d t h i s b o u n d .T h e o r e m 3 . F o r [ 3 = 1 , a n d j o b s j a n d k d e f i n e d s u c h t h a t o t j < o t k , i f tj < ot ~ a n d t , < e t k , t h e n T B > e l i.P r o o f . L e t r = a J e t k < 1 ( g i v e n ) ; x = t J c t j ; a n d y = t k / a k . B y d e f i n i t i on , 0 < x < 1 and 0 < y < 1 . E q . ( 9 )f o r T B m a y b e n o w r e - w r it t e n a s:

T B = M a j ,w h e r e

M - - 1 / ( 1 - r ) lo g { ( [ 1 - e x p ( - x ) ] [1 - e x p ( - y / r ) ] ) / ( [ l - e x p ( - y ) ] [1 - e x p ( - r x ) ] ) } . ( 1 2 )F i r s t , c o n s i d e r t h e v a l u e o f M a t r = 1 . S i n c e M b e c o m e s i n d e t e r m i n a t e , u s i n g L ' H o s p i t a l ' s r u l e w e o b t a i n :

L i m i ti n g v a lu e o f M ( a t r - - 1 ) = x / ( e x p ( x ) - l ) + y / ( e x p ( y ) - 1 ) = f ( x ) + f ( y ) . ( 1 3 )C o n s i d e r th e d e r iv a t i v e o f f ( x ) w i t h r e s p e c t t o x :

d f ( x ) / d x = f ' ( x ) = { ( e x p ( x ) - l ) - x e x p ( x ) } / ( e x p ( x ) - l ) 2 . ( l 4 )A t x = 0 , f ' ( x ) i s i n d e t e rm i n a t e . A p p l y i n g L ' H o s p i t a l ' s r u l e o n c e ag a i n , th e l im i t i n g v a l u e o f f ' ( 0 ) i s

n e g a t i v e ( = - 0 . 5 ) . F u r t h e r, f ( x ) i s c o n t i n u o u s i n x , a n d h a s n o s t a t io n a r y p o i n t ( i. e ., t h e r e e x is t s n o x > 0w h e r e f ' ( x ) = 0 ). H e n c e , f ' ( x ) i s a l w a y s n e g a t i v e . S i n c e 0 _< x _< 1 , t h i s i m p l i e s t h a t f ( x ) a t t a i n s i t s m i n i m u mva l u e a t x = 1 .

I n a s i m i l a r f a s h i o n , w e c a n s h o w t h a t f ( y ) a t t a in s it s m i n i m u m v a l u e a t y = 1 . S u b s t i t u t i n g x = 1 a n d y = 1i n E q . ( 1 3 ) , w e o b t a i n :

M = { 1 / ( e - 1 ) + 1 / ( e - 1 )} = 1 . 16 . ( 1 5 )T h u s M > 1 . 1 6 a t r = 1 . N e x t , w e o b s e r v e t h a t M = o o a t r = 0 , a n d t h a t d M / d r < 0 f o r al l r , x , a n d y (s e e

L e m m a 1 i n A p p e n d i x B . H e n c e , i t f o l l o w s t h a t t h e m i n i m u m v a l u e o f M i s o b t a i n e d a t t h e e x tr e m e p o i n t( r = 1 ) o f a f e a s i b l e s p a c e , d e f i n e d b y r < 1 .

T h e r e f o r e , T B = M c t y > 1 . 1 6 c t j > ct j . [ ]C o r o l l a r y 1 . F o r ~ 5 = 1 , a n d j o b s j a n d k d e f i n e d s u c h t h a t o t < e L , i t f o l l o w s t h a t i f t j < c t j , a n d t k < ~ k , i t i ss u f f i c i e n t t o h a v e T i < o t j f o r t h e s e q u e n c i n g r u l e e s t a b l i s h e d i n T h e o r e m 2 t o b e v a l id .

A s s t a t e d e a r li e r , w h e n [3 = 1 , ta i s t h e m e a n t i m e b e t w e e n m a c h i n e f a i lu r e s . T h e p r o c e s s i n g t i m e o f e a c h j o bm a y h e n c e b e a s s u m e d t o b e s m a l le r t h a n i t s c o r r e s p o n d i n g o t v a l u e . F o r a ll p r a c ti c a l p u r p o s e s , i t m a y b er e a s o n a b l e t o a s s u m e t h a t t h e s u m o f p r o c e s s i n g t i m e s o f a ll j o b s i n a p r o d u c t i o n r u n w o u l d n o t e x c e e d t h es m a l l e s t a v a l u e . T h e a s s u m p t i o n s i n C o r o l l a r y 1 a r e t h e re f o r e t ru e a n d T , ( s u m o f p r o c e s s i n g t i m e s o f a ll j o b sf r o m j o b 1 t o j o b i ) w i ll n e v e r e x c e e d o t j ( a n d h e n c e , t h e t h re s h o l d T B ) .

E x t e n d i n g t h e r es u l ts o f T h e o r e m s 2 a n d 3 t o ea c h p a i r o f j o b s ( j , k ) , i t f o l l o w s t h a t t h e o p t i m a l s e q u e n c es h o u l d p r o c e s s t h e s e j o b s i n n o n - i n c r e a s i n g o t o r d e r . F o r c o n v e n i e n c e , w e c a l l s u c h a s e q u e n c e t h e A - o p t i m a l


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28 0 A.K . Ch a k rav a rty, N . Ba la k r i sh na n / Eu ro pea n J o u rna l o f O pera t io na l Resea rc h 96 (1996) 27 4-2 88s e q u e n c e . I t i s e s p e c i a l l y i n t e r e s t i n g t o n o t e t h a t t h e i n d i v i d u a l j o b p r o c e s s i n g t i m e s d o n o t p l a y a p a r t i n t h es eq uenc ing ru le i t s e l f .3 . 2 . S i m u l a t i o n r e s u l t s f o r A - o p t i m a l j o b s e q u e n c i n g r u l e

W e n o w u s e s i m u l a t i o n t e s t s t o s h o w t h e v a l i d i t y a n d e f f e c t i v e n e s s o f t h e j o b s e q u e n c i n g r u l e d e v e l o p e da b o v e . A l l t e s ts a r e ru n o n a s e q u e n c e o f 5 j o b s . F o r c o n v e n i e n c e , j o b s 1 t h r o u g h 5 a r e a l w a y s n u m b e r e d s u c ht h a t eL i < u / , i < j . T he A - o p t i m a l s e q u e n c e i s t h e r e f o r e 5 , 4 , 3 , 2 , 1 .

I n a l l o u r s i m u l a t i o n t e s ts , th e s m a l l e s t o t v a l u e ( u , ) i s s e t t o 3 0. T h e r e m a i n i n g u i a r e t h e n r a n d o m l ys e l e c t e d o v e r s m a l l ( 3 0 - 3 5 ) , m e d i u m ( 3 0 - 5 0 ) , o r l a r g e ( 3 0 - 8 5 ) r a n g e s s u c h t h a t u ~ < u j , i < j . O n c e t h e u iv a l u e s a r e s e t , T5 ( = ~ = ~ t,) is v a r ie d o v e r v a l u e s f r o m l 0 t o 3 0 . O b s e r v e t h a t in k e e p i n g w i t h t h e e x p l a n at i o nf o l l o w i n g C o r o l l a r y l , t h e m a x i m u m T 5 v a l u e m a y n o t e x c e e d u i.

F o r e a c h T5 v a l u e , 1 0 0 d i f f e r e n t c o m b i n a t i o n s o f t i v a l u e s ( i = 1 t o 5 ) a r e r a n d o m l y g e n e r a t e d s u c h t h a te a c h t i ~ ( 1 , c t J 2 ) , a n d E ~ = l ti = T 5. F o r e a c h s e t o f t i v a l u e s , t h e M S P f o r a l l 1 2 0 ( = 5 ! ) s e q u e n c e s a r e f i r s tc a l c u l a t e d . T h e p r o b a b i l i t y d i f f e r e n c e s ( p e r c e n t a g e ) b e t w e e n t h e b e s t a n d w o r s t s e q u e n c e s , a n d b e t w e e n t h eA - o p t i m a l a n d w o r s t s e q u e n c e s , a r e th e n n o t e d .

I n F i g . 2 , th e p e r c e n t a g e d i f f e r e n c e s ( a v e r a g e o f 1 0 0 r u n s ) a r e s h o w n a t v a r i o u s T5 v a l u e s , f o r d i f f e r e n ts - r a n g e s . I t c a n b e s e e n t h a t f o r a l l T 5 v a l u e s s h o w n , t h e A - o p t i m a l s e q u e n c e i s a l w a y s o p t i m a l , a n d t h a t

o "

S m a l l a l p h a - r a n g e/

1 5 2 1 0 ; 5S u m o f p r o a e s s i n g t i m e s . . . . 3 0. .. ... . ( B e a t ) - ( W o r a t ) - - ( A -O p t im a l )- ( W or s t ) i

F i g . 2 . P e r c e n t a g e d i f f e r e n c e i n p r o b a b i l i t i e s ([ 3 = 1 ).


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A.K. Chakravarty, N. Balakrishnan European Journal of Operational Research 96 (1996) 274-288 281s i g n i f i c a n t i m p r o v e m e n t s i n M S P m a y b e a c h i e v e d b y u s i n g t h e A - o p t i m a l s e q u e n c e ( a s c o m p a r e d t o a r a n d o ms e q u e n c e ) . F u r t h e r , t h e s e b e n e f i t s i n c r e a s e w i t h b o t h T5 , and the rang e o f a va lues .

4 . S e q u e n c i n g r u l e s w i t h i n c r e as i n g f a i l u r e r a t e sT h e A - o p t i m a l s e q u e n c i n g r u l e a s s u m e s t h a t t h e m a c h i n e f a i l u r e r a t e d u e t o e a c h j o b i s c o n s t a n t o v e r t i m e .

H o w e v e r , i n ap p l i c a t i o n s s u c h a s t h o s e u s i n g m e c h a n i c a l c o m p o n e n t s , t h e s i g n i f i c a n t c h a n g e i n t h e s h a p e o f th eP D F f o r j o b s w i t h m a c h i n e f a i l u r e r a t e s th a t i n c r e a s e w i t h m a c h i n e - u s e t i m e ( i . e. , [3 > 1 ) w o u l d s t r o n g l y s u g g e s tt h a t t h e A - o p t i m a l r u l e m a y n o t r e m a i n v a l i d . W h e n 13 > 1 , t h e P D F c u r v e i n c r e a s e s f o r a w h i l e b e f o r ed e c r e a s i n g [ 2] . T h i s w o u l d i n d i c a t e t h a t a t l e a s t u p t o c e r ta i n v a l u e s o f p r o c e s s i n g t i m e s , t h e o p t i m a l s e q u e n c em a y b e j u s t t h e o p p o s i t e o f t h e A - o p t i m a l s e q u e n c e . I n T h e o r e m 4 b e l o w , w e f i r s t i n v e s t i g a t e t h i s p h e n o m e n o nf o r a s i t u a t i o n i n v o l v i n g o n l y 2 - j o b s .T h e o r e m 4 . I f 13 > 1, a n d j o b s j a n d k a r e d e f i n e d s u c h t h a t ~ t < o t k , t h en i t is s u f f ic i e n t t o h a v e t j + t k < T B( a n o n - n e g a t i v e t h r es h o l d ) f o r P M ( j , k ) > P M ( k , j ) , i .e ., t h e j o b s m u s t b e p r o c e s s e d i n n o n - d e c r e a s i n g o r d e r o fe t values .

P r o o f . E x p r e s s in g P M ( j , k ) an d P M ( k , j ) u s i n g E q . ( 5 ) , t h e f o l l o w i n g i n e q u a l i t y m u s t h o l d i f P M ( j , k ) > P M ( k , j ) :e x p ( - t J % ) ~ - e x p ( - t J % ) a + e x p ( - t k / % ) - e x p ( - t , / % ) ~ + e x p ( - [ tj + t k ] / % ) ~

- e x p ( - [ tj + t , ] / % ) ~ < 0 . ( 1 6 )L e t t i n g r = e t j / a k < 1 (g iven) ; x = t J o t f i y = t k / o t j ; a n d u = ( t j + t k ) / o t j = ( x + y ) , E q . ( 1 6 ) b e c o m e s

( fo r a g iv en r and [ 3) :{ e x p ( - r x ) ~ - e x p ( - x ) O + e x p ( - r y ) a - e x p ( - y ) ~ + e x p ( - u ) O - e x p ( - r u ) ~ }

= { f ( x ) + f ( y ) - f ( u ) } < 0 . ( 17 )O b s e r v e t h a t f ( x ) , f ( y ) , a nd f ( u ) are i d e n t i c a l f u n c t i o n s . C o n s i d e r t h e b e h a v i o r o f f ( t o ) w h e r e t o = x , y , o r

u . C l e a r l y , f ( t o ) i s c o n t i n u o u s a n d n o n - n e g a t i v e i n t o. F u r t h e r , f ( 0 ) = 0 a n d f ( ~ ) = 0 . N e x t , w e e x a m i n e t h ed e r i v a t i v e o f f ( t o ) w i t h r e s p e c t t o t o :d f ( o J ) / d t o = f ' ( to ) = /3 o~ t3- ' ( ex p( - to ) t3 _ r t~ex p( _ r to ) t~ ) . (18 )

F r o m E q . ( 1 8 ) , w e f in d t h a t f ( t o ) h a s n o n - n e g a t i v e s t a t io n a r y p o i n ts (i . e ., f ' ( t o ) = 0 ) a t t o = 0 a n d S , w h e r e :S = ( / 3 l o g ( 1 / r ) / ( 1 - r t3 ) ) i / t3 . (19 )

T h e e x i s t e n c e o f t h e s e s t a t i o n a r y p o i n t s i m p l i e s t h a t f ( t o ) h a s a p o i n t o f i n f le c t i o n ( d e n o t e d b y qb ) b e t w e e nto = 0 an d S ( i .e . , d 2 f ( d ? ) / d J = f " ( t o ) = 0 ) . H e n c e , f ' ( t o ) i s a n i n c r e a s i n g f u n c t i o n ( i . e . , f " ( t o ) > 0 ) b e t w e e nto = 0 and ~b , and f ' ( t o ) i s a dec rea s ing f unc t ion ( i . e . , f " ( to ) < 0 ) be tw een to = d~ and S .

C o ns id e r f ( u ) , u _< qb ( i . e . , t + t k i s b o u n d e d b y a n o n - n e g a t i v e m u l t i p l e o f a j ) . U n d e r t h i s c o n d i t i o n , itf o l l o w s f r o m L e m m a 2 in A p p e n d i x B ( b y se t ti n g th e v al u e o f " a " t o z e r o ) t h at f ( u ) > { f ( x ) + f ( y ) }. O b s e r v etha t u > ~b ma y n o t inva l ida te the a bov e ineq ua l i ty , and henc e u _< ~b i s no t a neces s a ry cond i t ion .

T h u s , P M ( j , k ) > P M ( k , j ) . [ 7A s i n S e c t i o n 3 , w e n e x t e x a m i n e w h e t h e r t h e a b o v e r u l e h o l d s e v e n w h e n t h e r e a r e s e v e r a l j o b s i n t h e

s e q u e n c e .


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282 A.K. Chakravarty, N. Balakrishnan European Journa l of O perational Research 96 (1996) 274 -28 8T h e o r e m 5 . I f 1 3 > 1 , a n d j o b s j a n d k a re d e f i n e d su c h t h a t e t < e t k , t h e n i t is su f fi c ie n t t o h a v eT + t j + t k < T B ( a n o n - n e g a t i v e t h re sh o l d ) f o r P M ( 1 . . . . . i , (j , k ) ,m . . . . . N ) > P M ( 1 . . . . . i , (k , j ) ,m . . . . . N )( w h e re T ,. is t h e su m o f p ro c e s s i n g t i m e s o f a l l j o b s f ro m j o b 1 t o j o b i ) .P r o o f . I f P M ( 1 . . . . . i , ( j , k ) . . . . . N ) > P M ( 1 . . . . . i , ( k , j ) . . . . . N ) , i t w o u l d f o l l o w t ha t :

e x p ( - r , / a D ~ - e x p ( - [ r ~ + t k ] / a k ) ~ + e x p ( - - I t , + t k ] / % ) ~ - - e x p ( - - [ r , + tk + t j ] / % ) ~> e x p ( - T ~ / % ) " - e x p ( - [ T , . + t i l l % ) " + e x p ( - [ T ~ + t s ] / a , ) " - e x p ( - [ T ~ + t j + , k ] / a , ) ~ .

( 2 0 )I f r = o L j / e tk < l ( g i v e n ) ; a = T i/o~ ;; x = t j / ~ j ; y = tk /e ~ ;; a nd u = ( t j + t k ) / o t ; = ( x + y ), E q . ( 2 0 )

b e c o m e s :ex p( - r [ a + x ]) t3 _ ex p( - [ a + x ])/~ + e xp ( - r [ a + y ] ) a - ex p( - [ a + y ] ) t3 + ex p( - [ a + u ] ) a

- e x p ( - r [ a + u ] ) / 3 + e x p ( - a ) t3 - e x p ( - r a ) # < 0 , ( 2 1 )w h i c h , f o r a g i v e n r a n d [ 3, m a y b e e x p r e s s e d a s :

f ( a + x ) + f ( a + y ) - f ( a + u ) - f ( a ) < 0 . ( 2 2 )

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b I O O I I ~ D f . . . . .. . . . . . . . . t m O I I Q I D D Q m Q m l I O D Q j D D D I ~ Q W m~ . .. . O ~m . .. . O ~D Oe l d IO ~O ~Q OI ~w O~ OD Og

l2 ; 0 2 5S u m o f p r o o e s s i n g t i m e s . . . .

3 0

. . . . . . . ( B e s t )- ( W o r st ) - - ( B - O p ti m a l ) -( W o r s t) IFig. 3. Percentage difference in probabilities (sma ll ~ range).


10/15

A.K . Chakravarty, N. Balakrishnan / European Journal of Operational Research 96 (1996) 2 74 -28 8 28 3O n c e a g a i n , w e n o t e t h a t t h e s e a r e i d e n t ic a l f u n c t i o n s . C o n s i d e r t h e b e h a v i o r o f f ( t o ) w h e r e t o - - - a + x ,

a + y , o r a + u . A s in T h e o r e m 4 , w e f i n d t h a t f ( o ~ ) h a s n o n - n e g a t i v e s t a t i o n a r y p o i n t s ( i .e . , f ' ( t o ) = 0 ) a tt o = 0 a n d S w h e r e t h e e x p r e s s i o n f o r S i s g i v e n i n E q . ( 1 9 ). S i n c e x a n d y a r e b o t h n o n - n e g a t i v e , i t f o l l o w st h a t a < S . H o w e v e r , a s s h o w n i n w h a t f o l l o w s , t h is c o n d i t i o n i s n e v e r b i n d i n g .

O n c e a g a i n , t h e e x i s t e n c e o f t h e s t a t io n a r y p o i n t s i m p l i e s t h a t f ( t o ) h a s a p o i n t o f in f l e c t io n ( d e n o t e d b y ~ b ),s u c h t h a t f ' ( t o ) i s a n i n c r e a s i n g f u n c t i o n ( i .e . , f " ( t o ) > 0 ) b e t w e e n t o = 0 a n d ~ b.

C o n s i d e r f ( a + u ) w h e r e a + u < ~b ( i .e . , t h e s u m o f p r o c e s s i n g t i m e s T , , t j, a n d t k i s b o u n d e d b y an o n - n e g a t i v e m u l t i p l e o f a t ) . T h i s i m p l i e s t h a t a _< S ( s in c e dp < S ) , a n d h e n c e t h e n e c e s s a r y c o n d i t i o n s t a te da b o v e f o r t h e e x i s t e n c e o f S i s a l w a y s s a t i s f i e d .

T h e r e f o r e , i f a + u < d ~, i t f o l l o w s f r o m L e m m a 2 i n A p p e n d i x B t h a t:f ( a ) + f ( a + u ) > _ f ( a + x ) + f ( a + y ) . ( 2 3 )

A s b e f o r e , a + u < ~b i s n o t a n e c e s s a r y c o n d i t i o n . H e n c e , P M ( 1 . . . . . i , ( j , k ) ,m . . . . . N ) >P M (1 . . . . . i , (k , j ) ,m . . . . . N ) i f TB = ~b, i .e. , (T + t j + tk ) /c t j 3 0

....... B e s t ) - ( W o r s t ) ~ ( B - O p t i m a l ) - ( W o r s t ) [

Fi g . 4 . P e rc e n t a g e d i f fe re n c e i n p ro b a b i l i ti e s (m e d i u m e t r a n g e ) .


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284 A.K. Chakravarty, N. Balakrishnan/ European Journal of Operational Research 96 (1996) 274-288F o r c o n v e n i e n c e , w e c a l l a s e q u e n c e o f jo b s a r r a n g e d i n n o n - d e c r e a s i n g e t o r d e r th e B-optimal s e q u e n c e . I n

w h a t f o l l o w s , w e u s e s i m u l a t i o n t e s t s t o s t u d y t h e b e h a v i o r o f T B w i t h r e s p e c t t o 1 3. F o r a g i v e n v a l u e o f 13, w ea l so s t u d y t h e p e r f o r m a n c e o f t h e B - o p t i m a l s e q u e n c e w h e n T > T B .4.1. Simulation results fo r B-optimal job sequencing rule

T h e s i m u l a t i o n t e st s a re d e s i g n e d t o a n s w e r t h e f o l l o w i n g q u e s t io n s : ( i ) h o w s u p e r i o r i s t h e m a k e s p a np r o b a b i l i t y o f t h e B - o p t i m a l s e q u e n c e o v e r th e w o r s t s e q u e n c e ? ; ( ii ) f o r T > T B ( i .e . , B - o p t i m a l se q u e n c e i s n o tn e c e s s a ri l y o p t i m a l ) , h o w i n f e ri o r i s th e B - o p t i m a l s e q u e n c e o n a v e r a g e a s c o m p a r e d t o t h e o p t i m a l s e q u e n c e ?;a n d ( i i i ) f o r g i v e n v a l u e s o f o t i , t i, a n d 13, u n t i l w h a t v a l u e o f T B d o e s t h e B - o p t i m a l s e q u e n c e alwaysm a x i m i z e t h e m a k e s p a n p r o b a b i l i t y ( if T < T B ) ?

O n c e a g a i n , t h e t e s t s a re r u n o n a s e q u e n c e o f 5 j o b s w h i c h a r e n u m b e r e d s u c h t h a t e t i < e tj , i < j . T h eB - o p t i m a l s e q u e n c e i s t h e r e f o r e 1 , 2 , 3 , 4 , 5 . T h e m e c h a n i c s o f t h e s e t e s t s a r e a s d e s c r i b e d i n S e c t i o n 3 , a n d o n c ea g a i n , th e p e r c e n t a g e m a k e s p a n p r o b a b i l it y d i f f e r e n c e s b e t w e e n t h e b e s t a n d w o r s t s e q u e n c e s , a n d b e t w e e n t h eB - o p t i m a l a n d w o r s t s e q u e n c e s , a r e c a l c u l a t e d a f t e r e a c h t e s t.

I n F i g . 3 , F i g . 4 , a n d F i g . 5 , t h e s e p e r c e n t a g e d i f f e r e n c e s ( a v e r a g e o f 1 0 0 r u n s ) a r e s h o w n f o r v a r i o u s v a l u e so f T5 . T h e v a l u e s o f a i r a n g e b e t w e e n 3 0 a n d 3 5 i n F i g . 3 , b e t w e e n 3 0 a n d 5 0 i n F i g . 4 , a n d b e t w e e n 3 0 a n d 8 5

2 5

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. ~ 1 . 5 -

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5-

01 5

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2 0 5S u m o f p r o o e s s i n g t i m e s . . . . >

3 0

I . . . . . . . ( B e s t ) - ( W o r s t ) ( B - O p t i m a l ) - ( W o r s t )Fig. 5. Percentagedifference n probabilities large ct range).


12/15

A.K . Ch a kra va r ty , N . B a la k r ish na n / Eu ro pea n J o u rna l o f O pera t io na l Resea rc h 96 (1996) 27 4-2 88 285

i n F ig . 5 . As in Sec t ion 3 , i t i s r ea sonab le t o a ssume tha t T 5 w i l l n o t e x c e e d M i n { a i} ( = a ~ ) . T h e r e f o r e , t h em a x i m u m T5 va lue i s once aga in se t a t 30 .F r o m t h e f i g u r e s , w e n o t e t h a t s i g n i f i c a n t i m p r o v e m e n t s i n M S P c a n b e a c h i e v e d b y u s i n g t h e B - o p t i m a l

s e q u e n c e ( f o r e x a m p l e , a s s e e n i n F i g . 5 , M S P o f t h e B - o p t i m a l s e q u e n c e m a y b e o v e r 2 0 % b e t t e r t h a n t h a t o ft he w o r s t s e q u e n c e i n s o m e c a se s ). T h e s e M S P i m p r o v e m e n t s b e c o m e m o r e p r o n o u n c e d a s e i t h er th e r a n g e o f ava lues i nc reases , o r t he 13 va lue de f in ing the shape o f t he Weibu l l f a i l u re d i s t r i bu t ion inc reases .

Fr om the f i gures , we a l so no te t ha t fo r a g iven 13 va lue , a s t he range o f a i i nc reases , t he va lue o f t het h r e s h o l d T B o n t h e s u m o f j o b p r o c e s s i n g t i m e s i n c r e a s e s ( re c o l l e c t t h a t T B = M a x { T5} such tha t t he B-op t ima ls e q u e n c e i s a l w a y s o p t i m a l ) . I n f a c t i f T5 < M i n { a i ) (a s i n o u r s i m u l a t i o n a n a l y s i s ), t h e f i g u r e s s h o w t h a t e x c e p tf o r s m a l l a - r a n g e s , t h e B - o p t i m a l s e q u e n c e i s e i th e r o p t i m a l o r c l o s e to o p t i m a l f o r a ll T5 ( i . e . , t h re sho ld T B i sc l o s e t o M i n { a ~ }). H e n c e , f r o m a p r a c t ic a l v i e w p o i n t , t h e u s e o f t h e B - o p t i m a l s e q u e n c e i s l i k e l y t o y i e l ds i g n i f i c a n t b e n e f i t s i n t e r m s o f m a x i m i z i n g m a k e s p a n p r o b a b i l i t i e s . F o r T5 > T B , t h e a v e r a g e m a k e s p a np r o b a b i l i t y d i f f e r e n c e b e t w e e n t h e B - o p t i m a l a n d b e s t s e q u e n c e s i n c r e a s e s r a p i d l y w i th T5.

5 . C o n c l u s i o nW e h a v e s h o w n c o n c l u s i v e l y t h a t j o b s e q u e n c i n g i s c ri t ic a l in m i n i m i z i n g t h e e x p e c t e d m a k e s p a n i n a

d e t e r i o r a t i n g f l e x i b l e m a c h i n e w h e n i t p r o c e s s e s m u l t i p l e j o b s , e a c h c o n t r i b u t i n g t o m a c h i n e d e t e r i o r a t i o n a t ad i f f e r e n t r a t e . O p t i m a l j o b s e q u e n c i n g r u l e s h a v e b e e n d e v e l o p e d f o r m a x i m i z i n g t h e s u r r o g a t e o b j e c t i v ef u n c t i o n , m a k e s p a n p r o b a b i l i t y , f o r s c e n a r i o s w h e r e t h e m a c h i n e f a i l u r e d i s t r i b u t i o n s d u e t o e a c h j o b a r e b o t he x p o n e n t i a l a n d w e i b u l l . S i m u l a t i o n t e s t s h a v e b e e n u s e d t o s h o w t h a t t h e s e j o b s e q u e n c i n g r u l e s m a k e i tp o s s i b l e t o s i g n i f i c a n t l y i m p r o v e t h e m a k e s p a n p r o b a b i l i t y ( a n d h e n c e t h e e x p e c t e d m a k e s p a n ) .

T h e d i f fe renc e in t he sequ enc ing ru l e s t ruc tu re s whe n 13-- 1 and 13 > l i s ve ry revea l ing , and m ay h o lds i g n i f ic a n t i n s i g h ts f o r m a n a g e r s . F o r e x a m p l e , t h e s y s t e m m a y s t a rt i n t h e 13 = 1 s t at e i m m e d i a t e l y a f t e r am a in tenanc e , and th en g rad ua l ly de t e r io ra t e ( a s a func t ion o f t ime ) t o t he 13 > 1 s t a te . Un de r su ch s i t ua t ions , i tm a y b e p o s s i b l e t o u s e t h e j o b s e q u e n c i n g r u l e s to a d d r e s s t h e f o l l o w i n g i s s u e s: ( i ) h o w s h o u l d j o b b a t c h e s b ef o r m e d s o a s t o b e s y n c h r o n i z e d w i th s c h e d u l e d m a i n t e n a n c e s ; ( i i) h o w l o n g a f t e r a m a i n t e n a n c e s h o u l d t h e r u l efo r 13 = 1 be u sed , b e fore s wi t ch ing to t he ru l e fo r 13 > 1 ; ( i i i) c an con t ro l cha r t s be con s t ruc t ed to t r ack the sh i f tf rom 13 = l t o 13 > 1? , and ( i v ) wha t i s t he imp ac t on ba t ch s i ze o f swi t ch ing f rom one ru l e t o an o the r?

W e v i e w t h i s p r o b l e m a s p a r t o f a la r g e r s e t o f i s s u e s i n v o l v i n g s e q u e n c i n g o f j o b s i n f le x i b l e m a c h i n e s .S i n c e m i n i m i z a t i o n o f e x p e c t e d m a k e s p a n r e q u i r e s j o b s t o b e p r o c e s s e d i n a s p e c i f ic o r d e r , s u c h a n o r d e r i n g o fj o b s m a y n o t e n a b l e f u l l e x p l o i ta t i o n o f a m a c h i n e ' s f l e x i b i li t y . A d d i t i o n a ll y , t h e j o b s e q u e n c e h a s t o c o n s i d e rn o t o n l y e x p e c t e d m a k e s p a n , b u t a l s o f a c t o r s s u c h a s h o l d i n g c o s t s a n d t o o l - m a g a z i n e c a p a c i t i e s . H e n c e , t h er u l e s d e v e l o p e d h e r e m u s t b e r e c o n c i l e d w i t h o t h e r s e q u e n c i n g r u l e s i n o r d e r t o o b t a i n a g l o b a l l y o p t i m a ls e q u e n c e .

Y e t a n o t h e r i s su e t h a t n e e d s t o b e a d d r e s s e d i s h o w t o r e s p o n d t o a c t u a l m a c h i n e f a i lu r e s . S p e c i f i c a l l y , i f o u rm o d e l , i n s t e a d o f a s s u m i n g a c o n s t a n t re p a i r ti m e a t e a c h m a c h i n e f a i l u r e , w e r e t o i n c o r p o r a t e th e e x p e c t e d c o s to f s u c h f a i l u r e s , h o w w o u l d t h e s t r u c t u r e o f t h e s e q u e n c i n g r u l e s b e i m p a c t e d ?

A p p e n d i x A . D e v e l o p m e n t o f t h e f a i lu r e p r o b a b i l it y P ( i , j )F i r s t , w e c o n s i d e r a p r o b l e m i n v o l v i n g t w o s i m i l a r job s i and j ( i . e . , a i = Or ) wi th d i f fe re n t p roces s ing

t i m e s t i a n d t j , r e s p e c ti v e l y . I f j o b i i s p r o c e s s e d f i r s t, t h e p r o b a b i l i t y o f m a c h i n e f a i lu r e b e t w e e n t i m e 0 ( w h e nthe job i s i n t roduced) and t ime t i i s g iven by :F i ( t i ) = f o ' f~ . ( t ) d t = 1 - e x p ( - t i / a i ) ~ ( A . 1 )


13/15

286 A.K . Ch a kra va r ty , N . B a la k r ish na n / Eu ro pea n J o u rna l o f O pera t io na l Resea rc h 96 (1996) 2 74 -28 8I f th e m a c h i n e s w i t c h e s i n s ta n t a n e o u s l y f r o m j o b i t o j o b j a t t im e t i, th e p r o b a b i l i t y o f m a c h i n e f a i l u r e

b e t w e e n t i m e t i and t i + t j g iv en t ha t t he mach ine i s ope ra t ing sa t i s fac to r i l y a t t ime t i, i s then:F j ( t i , t i + t j ) = [ 1 - F i ( t i ) ] - l ~ , " + ' J f j ( t ) d t = l - e x p ( [ t i / c t i ] ~ - ( [ t i + t j ] / a i ) ~ ) . ( A . 2 )

Ob se rve tha t i f 13 = 1 , due to t he me m ory - l e ss p rope r ty o f t he exponen t i a l d i s t r ibu t ion , Fj . ( t i , t i + t j ) i s thes a m e a s F j ( t j ) . N o w , P ( i , j ) m a y b e e x p r e s se d a s:

P ( i , j ) = 1 - e x p ( - [ t i + t j ] / o t i ) ~ , ( A . 3 )w h i c h i s i n d e p e n d e n t o f t h e o r d e r i n w h i c h t h e j o b s a r e p r o c e s s e d b y t h e m a c h i n e f o r a l l v a l u e s o f 1 3 .

N e x t , w e c o n s i d e r a p r o b l e m i n w h i c h j o b s i an d j a re d i f f e ren t , i . e. , a i :~ a j . I f job i i s p roc essed f i r s t, t hep r o b a b i l i t y o f m a c h i n e f a i l u r e b e t w e e n t i m e 0 a n d t i i s o n c e a g a i n g i v e n b y E q . ( A . 1 ) . T h e c h o i c e o f t h ec o n d i t i o n i n g e v e n t i n t he ca l cu la t ion o f F j ( t i , t i + t j ) i s , how eve r , unc lea r . A s sho wn in F ig . 1~ whi l e f j ( t ) i sre l evan t on ly be tw een t ime t / and t i + t j , i t is im pl i c i t l y a ssum ed to ex i s t f rom t ime 0 i t se l f. T he re fore , t hev a l i d i t y o f t h e m a c h i n e f a i l u r e P D F i s m a i n t a i n e d i f t h e c o n d i t i o n i n g e v e n t i s [ 1 - F j ( t i ) ] .

Ho we ve r , the use o f [1 - F j (t~) ] a s t he cond i t i on ing ev en t does n o t t ake in to acco unt t he fac t t ha t t he jobp r o c e s s e d b e t w e e n t i m e 0 a n d t i i s j o b i , a n d n o t j o b j . A s t h e m a c h i n e f a i lu r e P D F s w i t c h e s f r o m f ~ ( t ) t o f j ( t ) ,t h e r e w i l l b e a " c a r r y - f o r w a r d " o f m e m o r y r e g a r d i n g t h e w e a r o n t h e m a c h i n e d u e t o j o b i ( e v e n w h e n t h em a c h i n e f a i l u r e d i s tr i b u ti o n s d u e t o i n d i v i d u a l j o b s a r e e x p o n e n t i a l) .

W e i l l u s t r a t e t h i s p h e n o m e n o n u s i n g t h e f o l l o w i n g e x p e r i m e n t b a s e d o n t h e b a s i c p r i n c i p l e s o f p r o b a b i l i t y .C o n s i d e r K t r ia l s o f j o b i p r o c e s s e d o n t h e m a c h i n e ( f r o m t i m e 0 i n e a c h tr i al ) . L e t k~ d e n o t e t h e n u m b e r o ft h e s e tr ia l s in w h i c h t h e m a c h i n e b r e a k s d o w n b e f o r e t im e t~ ( c l e a r ly , k i < K ) . N o w c o n s i d e r K i n d e p e n d e n tt r ia l s o f j o b j p r o c e s s e d o n t h e m a c h i n e ( f r o m t i m e 0 i n e a c h t r ia l ). L e t k j i a n d k j2 d e n o t e , r e s p e c t i v e l y , t h en u m b e r o f t h e s e tr i al s i n w h i c h t h e m a c h i n e b r e a k s d o w n b e f o r e t i m e t i a n d b e t w e e n t i m e t i an d t i + t j (c l ea r ly ,k j i + k j 2 < K ) .W i t h r e s p e c t t o o u r p r o c e s s , t h e o n l y s a m p l e s p a c e s o f i n t er e s t i n t h e c o m p u t a t i o n o f P ( i , j ) , t h e probab i l i t yo f a t l e a st o n e m a c h i n e f a i l u r e b e f o r e t h e c o m p l e t i o n o f b o t h j o b s i a n d j ( p r o c e s s e d i n t h a t o r d e r ) , ar e t h e k im a c h i n e f a i lu r e s (w i t h j o b i ) b e t w e e n t i m e 0 a n d t i , and the k j ~ f a i lu r e s ( w i t h j o b j ) b e t w e e n t i m e t i an d t i + t j .H o w e v e r , i n o r d e r t o e n s u r e t h e v a l i d i t y o f P ( i , j ) , t h i s s a m p l e s p a c e n e e d s t o b e n o r m a l i z e d o v e r t h e e n t i r es a m p l e s p a c e o f t h e p r o c e s s . H e n c e , P ( i , j ) m a y b e e x p r e s s ed a s :

P ( i , j ) = ( k i + k j 2 ) / ( k i + k i 2 + [ K - k j l - k j 2 ] ) = ( k i + k j 2 ) / ( k i + K - k j i ) . ( A . 4 )E x p r e s s i n g t h i s i n t e r m s o f P D F s , w e o b t a i n :P ( i , j ) = ( f / ' f i ( t ) d t + f D " + t ' f j ( t ) d t ) / ( S o t ' f i ( t ) d t + S t 2 ( t ) d t ) . ( A . 5 )W e n o t e t h a t E q . ( A . 5 ) i m p l i c i t l y a s s u m e s t h a t f j ( t ) e x t e n d s f r o m t i m e t i u n t i l ~ . H o w e v e r , a s n o t e d e a r li e r ,

f j ( t ) i s r e l e v a n t o n l y b e t w e e n t i m e t i a n d t i + t j i n o u r p r o b l e m . H e n c e , t o g e n e r a l i z e P ( i , j ) fo r a l l jobs e q u e n c e s , w e a s s u m e t h a t a t th e c o m p l e t i o n o f th e l a s t r e a l j o b ( w h i c h m a y b e e i t h e r jo b i o r j o b j ) , t h em a c h i n e s w i t c h e s t o a d u m m y j o b ( i . e. , th e m a c h i n e f a i lu r e d i s tr i b u t io n b e t w e e n t i m e t i + tj and oo isc h a r a c t e r i z e d b y a w e i b u l l P D F w i t h a n a r b i t r a r y s c a l e p a r a m e t e r s o ) . H e n c e ,P ( i ' J ) = ( " f i ( t ) d t + ~ : ' + S f J ( t ) d t ) / ( S o' f i ( t ) . i + ' ~ )t + ~ ' f j ( t ) d t + f t, + i jf o ( t )d t . ( A . 6 )

O b s e r v e t h a t th e a c t u a l v a l u e o f P ( i , j ) e x p l i c i t ly d e p e n d s o n t h e v a l u e o f o t o a n d c a n t h e r e f o r e b e m a d e a sh i g h o r a s l o w a s d e s i r e d . H o w e v e r , w h e n P ( i , j ) i s u s e d o n l y f o r c o m p a r a t i v e p u r p o s e s o n l y , i . e . , t o c o m p a r e


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A.K. Chakravarty, N. Balakrishnan European Journal of Operational Research 96 (1996) 274-288 28 7the f a i lu re p roba b i l i ty o f one se qu e nc e wi th a no the r ( a s in a l l our a na ly s is ) , the va lue a ss igne d to a o be c o m e si r r e le va n t .

A p p e n d i x B . P r o o f s o f l e m m a sL e m m a 1 . I n T h e o r em 3 , f o r a l l v a lu e s o f r , x , a n d y , d M / d r < 0 ( w h e r e M g i v e n b y E q . ( 1 2 ) ) .

P r o o f . W e n o t e t h a t M i s c o n t i n u o u s a n d n o n - n e g a t i v e i n r ( th e l i m i t in g c a se w h e n r = 1 i s p r o v e d i n T h e o r e m3) . The de r iva t ive o f M wi th r e spe c t to r i s :d M 1 [ [ 1 - e x p ( - x ) ] [ 1 - e x p ( - y / r ) ] ] y e x p ( - y / r )d-"-~- = [ 1 r ]- - - - - ~ l g [1 ~ i [ ' 1 : ~ r - ' ~ - [ 1 - r ] - ~ [ 1 - - ' ~ x p ( - - y / r ) ]

x e xp( - r x )- [1 - r ] [ 1 - e x p ( - r x ) ] " ( B . 1 )

A t r = O, d M / d r < O. Fur the r , the re e xis t s no r (< 1 ) a t whic h r i M / d r = 0 ( i . e . , M d o e s n o t h a v e as ta t iona ry po in t in r ) . He nc e , d M / d r < 0 fo r a l l r , x , a nd y va lue s . [ ]L e m m a 2 . L e t f ( t o ) b e a c o n t i n u o u s n o n - n e g a t i v e f u n c t i o n w i t h a n i n c re a s in g n o n - n e g a t i v e s lo p e , i . e .,d 2 f ( t o ) / d t o 2 > 0 f o r a l l to . C o n si d e r t h e v a l ue o f f ( t o ) a t f o u r d i s t in c t p o i n ts : a , a + b , a + c , a n d a + b + c .T h e n ,

f ( a ) + f ( a + b + c ) > f ( a + b ) + f ( a + c ) . ( B . 2 )P r o o f . L e t O , ( > 0 ) a n d O 2 ( > 0 ) b e d e f in e d s u ch th a t { @ l ( a + b + c ) + @ 2 a = a + b } , w h e r e , - 1 " O 2 ~ - - 1.T h i s m a y b e s o l v e d t o o b ta i n 0 , = b / ( b + c ) a nd 2 = c / ( b + c) . S i n c e f ( t o ) i s c o n v e x ,

O l f ( a + b + c ) + 0 2 f ( a ) > f ( a + b ) . ( B . 3 )In a s imi la r ma nn e r , l e t ix l (> 0) a nd ix2 (> 0) be de f in e d suc h tha t {p . ,( a + b + c ) + p ,2 a = a + c } , whe reIx, + 1~2 = 1. Th is m ay be solv ed to o bta in I~1 = c / ( b + c) an d ~ x2 = b / ( b + c ). S i n c e f ( t o ) i s c o n v e x ,

i x , f ( a + b + c ) + i x2 f ( a ) > f ( a + c ) . ( B . 4 )A dd ing Eqs . (B .3) a nd (B .4) , we ge t F_ ,q. (B .2) . [ ]

R e f e r e n c e s[1] Barlow, R.E. , and Proschan F. , Mathematical Theory of Reliability, J ohn Wi le y a nd Sons , Ne w York, 1965 .[2] Bil l ington, R. , and Allan, R.N., Reliability Evaluation of Engineering Systems: Concepts and Techniques,Ple num Pre s s , Ne w York ,

1983.[3] B rowne , S . , a nd Ye c hia li , U . , "S c he d u l ing d e te r iora t ing j obs on a s ing le proc e s s or" , Operations Research 3 8 ( 1 9 9 0 ) 4 9 5 - 4 9 8 .[4] C ha k ra v a r ty , A.K. , a nd B a la k r is hna n , N. , " Im pa c t o f job- s e q ue nc e on the d own- t ime of a d e te r iorat ing fl e x ib le m a c h ine " , EuropeanJournal of Operational Research 8 1 ( 1 9 9 5 ) 2 9 9 - 3 1 5 .[5] C ha k ra v a r ty , A.K. , a nd L iu , J . , "O pt im a l j ob- re le a s e ord e r in a robot i c a s s e mbly c e l l " , Proceedings of the Third Special TIMS/ORSAConference on FMS, M IT, B os ton , M A , 1989 .[6] C hiu , W.K. , "Ec ono mi c d e s ig n of np-c ha r t s fo r p roc e s s e s s ub j e c t to a m ul t ip l ic i ty o f a s s ig na ble c a us e s " , Management Science 23

( 1 9 7 6 ) 4 0 4 - 4 1 1 .[7] Dhil lon, B.S. , Reliability Engineering in System Design and Operation, Va n No s t r and R e inhold , Ne w York , 1983.


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28 8 A.K. Chakravarty , N . Balakrishnan / Europ ean Journal o f Operational Research 96 (1996) 27 4-2 88[8] Groenvel t , H. , Pintelon, L. , and Seidmann, A. , "Product ion lot siz ing wi th machine breakdowns", Ma n a gement Sc ience 38 (1992)104-123.[9] Kan, S.H. , "M od el ing and software development qua l i ty" , I B M S y s te m s J o u rn a l 30 (1991) 351-362.[10] Kennedy, W.J. , Jr . , " Issue s in the maintenance of flexible manufacturing sys tem s", Ma intena nc e M a na gement In te rna t io na l 7 (1987)4 3 - 5 2 .[11] Lee, H.L. , and Rosenblat t, M .J., "E con om ic design and control of moni toring m echanisms in automated product ion sys tem s", l iE

Tra nsa c t io ns 20 (1988) 201-209.[12] McCal l , J . J . , "Maintenance pol icies for stochast ical ly fai l ing equipment : A survey" , M a n a g e m e n t S c i en c e 11 (1965) 493-524.[13] Meredi th, J .R. , "Au tom at ing the factory: Theory versus p ract ic e" , In te rna t io na l J o u rna l o f Pro du c t io n Resea rc h 25 (1987)1493-1510.[14] O'Connor, P. , P ra c t ic a l Re l ia b i li t y Eng ineer ing , Heyden Press, Philadelphia, PA, 1981.[15] Pierskal la, W .P. , and Voelker, J .A. , "A survey of maintenance m odels: The control and survei l lance of deteriorating system s", N a v a l

Resea rc h Lo g is t i c s Q u a r te r ly 23 (1976) 353-388.[16] Rosenblatt , M .J., and Lee , H.L. , "E con om ic production cycles wi th imperfect production pro cesse s" , l iE Transactions 18 (1986)4 8 - 5 5 .[17] Rosenblatt , M .J., and Le e, H.L., "A com parative study of continuous and periodic inspection policies in deteriorating productions y s t e m s " , l i e T ra nsa c t io ns 18 (1986) 2-9.[18] Smith, M.L. , Ramesh, R. , Dudek, R .A. , and Blair, E.L. , "Characteristics of U.S. flexible manufacturing systems - A S urv ey " , in:K.E. Stecke and R. Suri (eds.), P r o c e e d i n g s o f th e S e c o n d T I M S / O R S A C o n f er e n ce o n F M S , Elsevier, Amsterdam, 1986.[19] Special Issue on Reliability and Maintainability, O pera t io ns Resea rc h 32 (1984).[20] Stecke, K. , "Form ulat ions and solut ion of nonl inear integer planning problems for FM S" , Ma na gement Sc ienc e 29 (1988) 273-288.[21] Stecke, K . , and Talbot, F. , "Heu rist ic loading algorithms for flexible manufacturing syste m s", Pro c eed ings o f th e Sev enthInterna tional Conference on P rod uction Research, Windsor, Ontario, 1983.[22] Tang, C.S. , and Denardo, E.V. , "M od els arising from a flexible manufacturing m achine, Part I : Minimizat ion o f the number o f toolswi t ches" , O pera t io ns Resea rc h 36 (1988) 767-777.[23] Tang, C.S. , and Denardo, E.V. , "Models arising from a flexible manufacturing machine, Part I I : Minimizat ion of the number ofswitching instants" , O pera t io ns Resea rc h 3 6 (1988) 778-784 .[24] Vineyard, M .L. , and M eredi th, J .R. , "E ffe ct of maintenance pol icies on FM S fai lures " , In te rna t io na l J o u rna l o f P ro du c t io n Resea rc h30 (1992) 2647-2657.

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