multiobjective optimization in linear repetitive project scheduling

8/12/2019 Multiobjective Optimization in Linear Repetitive Project Scheduling

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ETtt )~e tp q ot ctm] Ep ea)vc t Op era t i o n a l Rese a rch . A n In t e rn a t i o n a l Jo u m al . V o l .6 , N o 3 (2 0 0 6 ) , p p .2 5 5 -2 6 9

M u l t i o b j e c t i v e O p t i m i z a t i o nin L i n e a r R e p e t it iv e P r o j e c t S c h e d u l in g

P a n d e l is G I p s i la n d i sProfe ssor, Department o f Pro ject Managem ent, Technological Educat ional Insti tute,4111 0 Lar issa, Greece.

b s t r a c tThe Cr i t ical Path Method (CPM) and the Repet i t ive Schedul ing Method (RSM) are the mostoften use d tools for the planning, scheduling and control Linea r Repeti t ive Proje cts (LRPs ).CP M focuses mo st ly on projec t s duration and cr i tical activi ties , wh i le RS M focuses onresou rce continuity. In this pap er w e present a l inear programm ing approa ch to address themu lt i obje ct ive nature o f dec isions construct ion managers face in scheduling LR Ps. The Mu lt iOb ject ive Linear Program ming m odel (MO LP -LR P) i s a parametric mo del that can optimize aschedule in terms o f duration , work-breaks, uni t com plet ion time and respect ive costs , whi le atthe sam e t ime the LP range sensit ivi ty analysis can provid e useful information regarding cos tt radeoffs between delay , work-break and uni t del ivery costs . MOLPS-LRP can generateal ternative schedules ba sed on the relative m agnitude and im portance o f different cos telements. In this sense i t provides managers with the capabil i ty to consider al ternat iveschedules besides those def ined by minimum durat ion (CPM) or minimum resource work-breaks (RSM). Demonst rat ive resul t s and analysis are provided through a wel l known in theliterature case study example.K ey w ord s: Linear Projects , Linear Programming, Scheduling1 I n t r o d u c t i o nL i n e a r R e p e t i t iv e c o n s t r u c t io n P r o j e c t s ( L R P s ) c o n s i s t o f a se t o f a c t i v it ie s t h a t a r er e p e a t e d s e q u e n t i a l ly a t d i f f e re n t l o c a t i o n s o r p r o j e c t u n i ts . T h e a c t i v it i e s f o l l o w al o g i c a l a n d t e c h n o l o g i c a l d r i v e n s e q u e n c e d e s c r i b e d b y t i m e o r d i s t a n c e c o n s t r a i n t sf o r t h e e n ti re l i fe s p a n o f th e p r o j e c t [ K a l l a n tz i s a n d L a m b r o p o u l o s ( 2 0 0 4 ) ] . B e c a u s eL R P r e s o u r c e s a r e u s e d i n a s e q u e n t i a l m a n n e r , e f f e c t i v e r e s o u r c e m a n a g e m e n t i sv e r y i m p o r t a n t b o t h i n t e r m s o f p r o j e c t c o s t a n d d u r a ti o n . A l s o , b e c a u s e o f t h ed i v i s i o n o f t h e p r o j e c t to i n d i v i d u a l u n i ts m e e t i n g i n t e rm e d i a te u n i t d e l i v e r y t i m e s i sa n o t h e r i m p o r t a n t i ss u e i n L R P s .T h e C r i ti c a l P a th M e t h o d ( C P M ) i s m o s t o f t e n u s e d f o r t h e p l a n n in g , s c h e d u l i n g a n dc o n t r o l o f s u c h p r o je c t s . N o n e t h e l e s s , n e t w o r k a n a l y s i s t e c h n i q u e s a r e d u r a t io n

1 T h is p a p e r i s p a r t o f a r e s e a r c h p r o j e c t o f t h e A R C H I M E D E S p r o g r a m m e o f t h e O p e r a t io n a lPro gram me fo r Educa t ion and ln i t i a l Voca t iona l Tra in ing in Gree ce under the 3 r Com mon Suppor t


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5 6 O p e r a t i o n a l R e s e a r c h . A n I n t e r n a t i o n a l J o u r n a l / V o l . 6 N o . 3 / S e p t e m b e r - D e c e m b e r 2 0 0 6

o r i e n te d a n d c a n n o t s u f fi c ie n t ly a d d re s s r e s o u r c e m a n a g e m e n t i ss u es . O t h e r m o r es u i t ab l e t e ch n i q u es t h a t f o cu s o n r e s o u rce u s ag e , s u ch a s t h e Rep e t i t i v e Sch ed u l i n gM e t h o d R S M ) h a v e b e e n d e v e l o p e d f o r s c h e d u l i n g a n d c o n t r o l li n g r e p e ti ti v epro jec t s .H o w e v e r t h e co m p l ex n a t u re o f r ep e t i t i v e p ro j ec t s ca ll s f o r d ec i s i o n s th a t c an n o t b eb a s e d s o l e ly o n t i m e o r r e so u r c e m a n a g e m e n t b u t r e q u ir e a m o r e h o l is t ic a p p r o a ch . I nt h is p ap e r w e ex p l o re t h e m u l t i o b j ec t i v e n a t u re o f d ec i s i o n m ak i n g i n r ep e ti t iv ec o n s t r u c ti o n p r o j e c t s a n d p r e s e n t a l i n e a r p r o g r a m m i n g p a r a m e t r i c m o d e l f o r m u l a t io nfo r s u p p o r t i n g t h e d e c i s i o n s o f co n s t ru c t i o n m an ag e r s . T h e r e s t o f t h e p ap e r i so rg an i zed a s fo l l o w s : T h e fo l l o w i n g s ec t i o n r e f e r s to d e f i n i t io n s , l it e r at u r e r ev i ew an da c lass i f i ca t ion o f l inear / repe t i t ive p ro jec t s . I t a l so inc ludes a c r i t i ca l p resen ta t ion o ft h e R S M . I n s e c t i o n 3 w e p r o p o s e a L i n e a r P r o g r a m m i n g f o r m u l a t io n fo r m o d e l l i n gL R P s , w h i c h i s d e s i g n e d i n a p a r a m e t r i c w a y s u c h a s t o a l l o w s i n g l e o r m u l t i p l eo b j ec t i v e o p t i mi za t i o n r eg a rd i n g t i me o r co s t e l emen t s o f t h e p ro j ec t . Sec t i o n 4f o l lo w s w i t h d e m o n s t r a t iv e r es u lt s b a s e d o n t h e a p p l ic a t io n o f t h e L P m o d e l t o a w e l lk n o w n l it e r a tu r e ex am p l e u n d e r d i f f e r en t s et s o f a s s u m p t i o n s an d p a rame t e r s . F i n a l l y ,s ec t i o n 5 co n t a i n s t h e co n c l u s i o n s an d p ro p o s e d d i r ec t io n s o f fu t u r e r e s ea rch .2 . L i n e a r Re p et i t i v e P r o j e ct sD u r i n g t h e l a s t d ecad es v a r i o u s t e rms h av e b een u s ed i n t h e l i t e r a t u r e , t o d e s c r i b em a i n l y c o n s t r u c t io n p r o je c t s th a t c a n b e d i v i d e d i n a n u m b e r o f u n i ts c o n s i s ti n g o f th es am e p ro j ec t a c t iv i t ie s , w h i ch a f t e r t h e i r co m p l e t i o n in o n e u n i t a r e r ep ea t ed i n t h en e x t o n e [ M a tt il a a n d A b r a h a m 1 9 9 8) ]. T h e m o s t p o p u l a r a m o n g t h e m is L in e a r a n dRep e t i ti v e P ro j ec t s L RP s ) , w h e re t h e t e rm l i n ea r r e f e rs t o co n s t ru c t i o n p ro j ec t s t h eac t iv i ti e s o f w h i c h a r e r ep ea t ed co n t i n u o u s l y in a h o r i zo n t a l f l o w , s u ch a s i n t h e ca s eo f s eg m en t s o f h i g h w ay s , b r i d g es , t u n n e l s , r a il w ay s , p i p e l i n e s , s ew er s , e t c. w h i l e t h et e rm r ep e t i ti v e d es c r i b e s co n s t ru c t i o n p ro j ec t s w h e re ac t iv i t ie s a r e r ep ea t ed i n d i s c r e t er ep ea t i n g u n it s a s i n th e ca s e h i g h - r i s e an d m u l t i s to ry b u i l d i n g s , m u l t i - h o u s i n gp ro j ec t s, e t c. m an y t i m es i n a v e r t i c a l d i rec t i o n .H o w e v e r th e n o t i o n o f L R P c a n b e e x t e n d e d b e y o n d th e c o n s t r u c t i o n in d u s t ry s in c eo t h e r t y p es o f p ro j ec t s can b e c l a s s i f i ed a s L R Ps a s t h ey h av e s i mi l a r ch a rac te r i st ic s .T y p i c a l e x a m p l e s a r e r e e n g i n e e r i n g p r o je c t s a n d p r o j e c t s i n v o l v i n g t h ei m p l e m e n t a ti o n o f n e w o r u p g r a d in g o f c o m p u t i n g sy s te m s w h e r e t h e s a m e t as k s a r et ak i n g p l ace s eq u en t i a l l y t o a s e r ie s o f b r an ch es o r u n i ts o f la rg e o rg an i za t i o n s a t h es am e o r d i f f e r en t g eo g rap h i ca l l o ca ti o n s .D i f f e r e n t m e t h o d s h a v e b e e n p r o p o s e d f o r p l a n n i n g , s c h e d u l i n g a n d c o n t r o l l i n g t h ec o n s t ru c t io n p r o c e s s o f t h e se t y p e s o f p r o j e c ts . T h e d e v e l o p m e n t o f R e p e t i ti v eS c h e d u l i n g M e t h o d R S M ) b y H a r ri s a n d I o a n n o u 1 9 9 8 ) a n d Y a n g 2 0 0 2 a) w h i c hcan b e i mp l emen t fo r b o t h p ro j ec t c a t eg o r i e s , i s t h e mi l e s t o n e fo r c l a s s i fy i n g t h e s ep ro j ec t s in t o a u n i q u e ca t eg o ry a s L RP s .


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2 1 Scheduling methodologies for LRPsTh e m o s t f r eq u en t ly u sed m e th o d f o r d e s ig n , p l an n in g , sch ed u l in g an d co n t r o l o fco n s t r u c t io n p r o j ec t s i s t h e Cr i t i c a l Pa th Me th o d ( CPM) [ Ma t t i l a an d Ab r ah am ,(1998) ] . Never the less , s ince the seven t ies researchers have been cha l leng ing i t sapp l icab i l i ty in an a t tempt to p rove i t s inadequac ies , whi le a t the same t ime proposeo th e r ap p r o ach es an d m e th o d s m o r e su i t ab l e f o r t h e co n s t r u c t io n p r o cess [ Pee r(1974) ; Ca r r and M ay er (1974) ; Dres s le r 1974; O B r ien (1975) ; A sh le y (1980);Se l inger (1980) ; Bi r re l l (1980) ; Johnston (1981) ; S t rada l and Cacha (1982) ; Russe l lan d Case l to n ( 1 9 8 8 ) ; Red a ( 1 9 9 0 ) ; Ha r m e l in k an d Ro win g s ( 1 9 9 8 ) ; Ha r r i s an dI o an n o u ( 1 99 8)] . Th e m a in r ea so n s th a t i n s t ig a t ed th e d ev e lo p m en t o f n ewap p r o ach es wer e th e in su f f i c i en cy o f n e two r k an a ly s i s t o d e sc r ib e th e r ep e t i t i v en a tu r e o f th e co n s t r u c tio n p ro cess an d to a l lo ca t e an d p l an n in g th e wo r k in th e v a r io u sconst ruc t ion si tes, i ts w eak ness to p rov id e un in te r rup ted u t i l iza t ion (w ork con t inu i ty )o f r e so u r ces ( wo r k t eam s) an d m in im iza t io n o f t h e id l e tim e , an d a l so th e weak n ess tom o d e l t h e l ea r n in g e f f ec t f r o m th e wo r k r ep e t it i o n . Mo r eo v e r , th e a r b i tr a r y e s t im a t io no f ac t iv i t ie s d u r a t io n in s t ead o f t h e e s tim a t io n o f p ro d u c t io n ra t e, t h e l a r g e n u m b er o fac t iv i t ies tha t a re requ i red in the ne twork fo r la rge and compl ica te p ro jec ts and thel ack o f i n f o r m a t io n co n ta in ed in th e d i ag r am a r e a l so co n s id e r ed s ig n i f i can twea k n esse s o f t h e CP M [ Bi rr e ll (1 9 8 0 ], s in ce em p h as i s i s g iv en to o p t im iz in g p r o j ec tco m p le t io n t im e an d n o t i n to th e o p t im iza tio n an d u t i l iz a t io n o f th e r e so u r ces an d th eco s t co n t r o l , wh ich i s t h e m a in co n ce r n wi th g r ea t im p o r t an ce f o r co n s t r u c t io ncontractors.Th e a l t e r n a t iv e ap p r o ach es th a t h av e b een d ev e lo p ed f o r LRPs a r e b ased m a in ly o ng r ap h ica l m e th o d s o n X - Y d i ag r am s , wh e r e o n e ax i s rep r e sen t s t im e an d th e o th e rw o r k p r o g re ss . Th ese m e th o d s can b e o r g an ized , acco r d in g to Ma t t i l a an d Ab r ah am .(1998) , in th ree bas ic ca tego r ies : i) those tha t a re based on the Lin e-of -B alanc e (LOB )techn ique , app l ied main ly to d iscre te p ro jec ts , i i ) those tha t a re based on LinearSch ed u l in g Me th o d ( LSM) , m o r e ap p r o p r i a t e f o r co n t in u o u s p r o j ec t s an d i i i ) t h o seth a t co m b in e th e p r ev io u s t ech n iq u es w i th o th e r s su ch a s d y n am ic p r o g r am m in g ,s to ch as ti c p r o g r am m in g , l i n ea r p r o g r am m in g , s im u la t io n e t c .W h i l e , a ll o f t h e se tech n iq u es ach iev ed to o v e r co m e th e in ad eq u ac ie s o f PER T/CP M ,th ey f a i led to o b ta in w id e accep tan ce an d p r ac t i ca l u se . Th e m a in r ea so n s a r e b ecau se ,th ey h av e b een m o r e co m p l i ca t ed th an th ey sh o u ld h av e b een wh i l e t h e r e i s n o t anaccep tab le a lg or i thm (process ) fo r iden t i fy ing the p ro jec t s c r i t ica l pa th an dsu b seq u en t ly th e p r o j ec t co m p le t io n t im e [ Har ri s an d I o an n o u ( 1 9 9 8) ; H a r m e l in k an dR ow ings (1998); Ka l lan tz is and Lam bropo ulos , (2003) ]. The lack o f a c r i tica l pa thiden t i f ica t ion a lgor i thm can be a t t r ibu ted to the fac t tha t in a l l these methods theunder ly ing assumpt ions i s tha t a l l ac t iv i t ies must considered as c r i t ica l , fo r be t te rcon t ro l o f a p ro jec t [P eer (1974)] . S ign i f ican t e f fo r t s how eve r have be en m ade the las tf ew y ea r s t o war d s th e d ev e lo p m en t o f an accep tab le an d accu r a t e a lg o r i th m to


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58 Operational Research. An International Journal / Vol.6 No.3 / September - December 2006

d e t e r m i n e th e c r i ti c a l p a t h i n L R P s [ H a r m e l i n k a n d R o w i n g s ( 1 9 9 8 ); H a r m e l i n k(2 0 0 1 ) ; Ma t t il a an d P a rk (2 0 0 3 ); K a l l an t z i s an d L am b ro p o u l o s (2 0 0 3 ) ; K a l l an tz i s an dL a m b r o p o u l o s ( 2 00 4 ) ].T h e R e p e t i ti v e S c h e d u l i n g m e t h o d ( R S M ) w a s i n t r o d u c e d b y H a r ri s a n d I o a n n o u( 1 99 8 ), a n d w a s f u r t h e r d e v e l o p e d w i t h w o r k t h at f o l lo w e d [ Y a n g a n d I o a n n o u(2 0 0 1 ); Y an g (2 0 0 2 a ) ; Y an g (2 0 0 2 b ) ; I o an n o u an d H ar r is ( 2 0 0 3 ) ; I o an n o u an d Y a n g(2 0 0 3 ); I o a n n o u an d Y an g (2 0 0 4 ) ; Y an g an d Io an n o u (2 0 0 4 )] . I t is p r im ar i l y ag rap h i ca l r ep re s en t a t i o n o f th e p ro j ec t o n an X -Y d i ag ram an d i ts o b j ec t i v e i s t oi n t eg ra te ex i s ti n g me t h o d s i n t o a g en e ra l i zed o n e t h a t en s u re co n t i n u o u s r e s o u rceu t il iz a ti o n. R S M c a n b e u s e d f o r t h e s c h e d u l i n g o f b o t h d i s c r e te a n d c o n t i n u o u sp ro j ec ts . Fo r d i s c r e t e p ro j ec t t h e r ep ea t ed u n i t s (w o rk p ro g re s s ) a r e u s u a l l y d r aw n i nt h e Y ax i s an d t h e e l ap s ed p ro j ec t t i me o n t h e X ax i s is t h e t i me , w h i l e fo r co n t i n u o u sp ro j ec t s th e t i me i s d r aw n o n t h e Y ax i s an d t h e r ep ea t ed u n i t s i n x ax i s.R S M u s e s a p u l l - s y s t e m a p p r o a c h , w h e r e t h e f i n is h - ti m e o f t h e p r e d e c e s s o r a c t iv i ty i sp u l l ed fo rw ard i n o rd e r to m ee t t h e s t a r t d a t e o f t h e s u cces s o r i n o rd e r t o ach i ev ew o rk co n t i n u i t y an d u n i n t e r ru p t ed r e s o u rce u t i li z a ti o n , in co n t r a s t w i t h t h e CP Mp u s h - s y s t e m , w h e r e t h e s t ar t o f e v e r y a c t iv i ty i s p u s h e d i n t i m e t o m a i n ta i n t h ep reced en ce r e l a t i o n s h i p s w i t h i t s p r ed eces s o r s . T h e o b j ec t i v e i n RSM i s n o t t h em i n i m i z a t i o n o f t h e p r o j e c t c o m p l e t i o n t i m e b u t a c h i e v i n g w o r k c o n t i n u i t y w h i c hl ead s to m i n i m i z i n g t h e o v e ra l l p ro j ec t co s t . I n co n s t ru c t i o n p ro j ec t s, t h em i n i m i z a t i o n o f t h e c o s t m a y b e m o r e d e s ir a b le t h a n t h e r e d u c t i o n o f t h e p r o je c td u ra t i o n [Y an g (2 0 0 2 b )] .T h e R S M a l g o r it h m h a s b e e n p r o g r a m m e d i n to c o m p u t e r i m p l e m e n t at io n , t h eRep e t i ti v e P ro j ec t P l an n e r (RP2 ) [Y an g (2 0 0 2 a )] . T h e a l g o r i t h m i n v o l v es tw o s t ag es :T h e f ir s t s t ag e i s s i mi l a r t o t h e fo rw ard p as s co m p u t a t i o n s o f CP M an d r e s u l ts i n th ec o m p u t a t i o n m i n i m u m p r o j e c t d u ra t io n . I n t h e s e c o n d s t a g e e a c h c o n t i n u i tyr e l a t io n s h i p p u l l s t h e p r ed ece s s o r s t o e l i mi n a t e t h e t i me g ap w i t h t h e su cces s o r t oen s u re w o rk co n t i n u i t y .A l t h o u g h RSM o p t i mi zes fo r co n t i n u i t y i t s an a l y s i s f ea t u r e s a r e l i mi t ed . I t c an o n l ya l l o w t r ad e o f f s b e t w een t i me g ap s an d p ro j ec t d u ra t i o n o n a t r i a l an d e r ro r b a s i s .Co s t co n s i d e ra t i o n s an d o t h e r co n t ro l v a r i ab l e s a r e n o t t ak en d i r ec t l y i n t oco n s i d e ra t i o n in t h e co m p u t a t i o n , b u t o n l y a s b ack -en d ca l cu l a t io n s .


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P Ipsilandis / M ultiobjective Optimization in Linear Rep etitive Project Scheduling 59

3 M u l t i Ob jecti ve L P Schedu l i ng o r L RPs3.1 Decision variables in scheduling LRPsS c h e d u l i n g o f L R P s i s i n p r a c t i c e m o r e c o m p l i c a t e d a n d r e l e v a n t d e c i s i o n s c o u l di n v o l v e m o r e c o n t r o l v a r ia b l e s t h a n j u s t d u r a t io n ( C P M ) o r r e s o u r c e c o n t i n u i t y(RSM) . T h e fo l l o w i n g l i s t i n c l u d es c r i t e r i a t h a t a r e i mp o r t an t t o co n s t ru c t i o nm a n a g e r s i n th e i r d e c i s i o n m a k i n g r e g a r d in g s c h e d u l i n g o f L R P si D u r a t i o n : Pro j ec t d u ra t i o n i s a k ey v a r i ab l e t o an y p ro j ec t s i n ce i t a f f ec t s t h e

p ro j ec t s f i n an c i a l p e r fo rm an c e ( co s t o v e r ru n s , p en a l t ie s ) an d r ep u t a t i o n o f th ep ro j ec t o rg an i za t i o n

ii. R e s o u r c e d e l a y : V i o l a t i n g th e w o r k c o n t i n u i ty o f th e s a m e t a s k b e t w e e ns u cces s i v e p ro j ec t u n i t s i n t ro d u ce s w o rk -g a p s t h a t i n c rea s e t h e co s t o f th e p ro j ec tb ecau s e o f i d l e re s o u rces .

iii. U n i t c o m p l e t i o n t i m e : C o m p l e t i o n o f t h e w o r k o n a p r o j e c t u n i t a f f e ct s th ep ro j ec t d e l iv e rab l e s an d i t co u l d h av e s i g n i f i can t f i n an c i a l im p l i ca t i o n s s i n ce th ep r o j e c t s c a s h r e c e i p ts d e p e n d o n c o m p l e t i n g in t e r m e d i a te d e l i v e ra b l e s. I n o t h e rc a se s , t h e c o m p l e t e d u n it s o f t h e p r o j e c t c a n b e c o m e o p e r a t i o n a l b e f o r e t h eco m p l e t i o n o f t h e en t i r e p ro j ec t t h u s r e s u l t i n g i n ea r l i e r c a s h i n f l o w s .

iv. S l a c k t i m e : Red u c i n g ac t i v i t y s l ack t i me i n t ro d u ces h i g h e r r i s k t o t h e p ro j ec t ,r eg a rd i n g u n i t co m p l e t i o n t i me an d o v e ra l l p ro j ec t d u ra ti o n .

V. N u m b e r o f u n i t s t h e p ro j ec t is d i v i d ed i n t o : E ach u n i t in t ro d u ces a ce r t a i n f i x edco s t f o r ma i n t a i n i n g a co n s t ru c t i o n s it e w h i c h i n c rea s e s t h e co s t o f t h e p ro j ec t b u ta t t h e s ame t i me s p l i t s t h e p ro j ec t i n t o s ma l l e r d e l i v e rab l e s w h i ch co u l d i mp ro v et h e p ro j ec t c a s h f l o w .

Fu r t h e rmo re s ch ed u l i n g d ec i s i o n s a r e r a r e l y b as ed o n l y o n an y s i n g l e v a r i ab l e .A l t e rn a t i v e p ro j ec t s ch ed u l e s , co mp ar i s o n s an d co s t t r ad eo f f s a r e o f t en n eed ed t oa r r i v e a t an accep t ab le o r o p t i m u m p ro j ec t s ch ed u l e . I n th i s a s p ec t t h e s ch e d u l i n gp r o b l e m c a n b e s e e n a s a m u l t i o b j e c ti v e p r o b l e m t h a t c a n b e a d d r e s s e d b y l in e a rp r o g r a m m i n g t e ch n iq u e s .T h e f o l lo w i n g li n ea r p r o g r a m m i n g f o r m u l a t i o n o f L R P s c h e d u l i n g c r e a te s a m o d e lw h i ch h a s a p a ram e t r i c s t ru c tu re t h a t c an ad d re s s s o m e o f t h e c r it e r ia ( i n d i v i d u a l l y o rc o m b i n e d ) r a i se d a b o v e .3.2 Formal description of the MO LP S R P m o d elC o n s i d e r a n L R P t h a t c o n s is t s o f a s e t o f m t a s k s r e p r e se n t e d b y t h e n o d e s o f a g r a p ha n d a s e t o f P d e p e n d e n c y re l a ti o n s h ip s th a t c a n b e r e p r e s e n t e d a s t h e a r c s o f t h eg r a p h . T h e s e d e p e n d e n c i e s c a n b e o f a n y ty p e o f t h e k n o w n o f p r o j e c t d e p e n d e n c i e s(SS , FS , SF , FF , w i t h o r w i t h o u t ti me- l ag b e t w een t h e d ep en d en t ac t iv i ti e s ). T h ep ro j ec t i s d i v i d ed i n t o n s ep a ra t e u n i t s i n a l i n ea r w ay w h e re w i t h o u t l o s s o fg en e ra l i t y t h e fo l l o w i n g g en e ra l a s s u m p t i o n s h o l d i n t h e m o s t pa r t:


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i. A l l t a s k s a r e p e r f o r m e d i n a l l u n i t sii. A t a s k c a n n o t b e p e r f o r m e d in a n y p r o j e c t u n i t b e f o r e th e s a m e t a s k i s c o m p l e t e d

i n th e p r e v i o u s u n i tiii . T h e s e t o f d e p e n d e n c i e s r e m a i n t h e s a m e i n all u n it s.Y a n g , 2 0 0 2 a ) l is t s a s e t o f p ra c t ic a l c o n c e r n s i n s c h e d u l i n g r e p e t it i v e p ro j e c t s t h a ta re s p e c ia l e x c e p t i o n s o f th e g e n e r a l a s s u m p t i o n s m a d e a b o v e a n d c a n b e e a s i lyh a n d l e d i n t h e L P f o r m u l a t i o n t h a t f o l l o w s .B a s e d o n t h e f o r m a l d e f i n i t io n g iv e n a b o v e , t h e L R P p r o b l e m c a n b e f o r m u l a t e d a s al in e a r p r o g r a m m i n g m o d e l a s f o l l o w s :M o d e l v a r ia b l e s a n d p a r a m e t e r sL e t i= 1 , 2 , . . . , m d e n o t e t h e p r o j e c t ta s k s a n d j = l , 2 , . . . , n t h e p r o j e c t u n i t s .D e f i n e : d o t h e d u r a t i o n o f t a sk i i n u n i t j a l t er n a t iv e l y i t c an b e f o r m u l a t e d a s

t h e a m o u n t o f t he c o r r e s p o n d i n g w o r k w i/ d i v i d e d b y t h e p r o d u c t i o nra te p iQ

s o , f i , t h e s ta r t a n d f i n is h t i m e r e s p e c t i v e l y o f ta s k i i n u n i t jP i t h e s e t o f p r e d e c e s s o r a c t i v it i e s t o t a s k iE t h e s e t o f a ll a c t i v it ie s w i t h o u t s u c c e s s o r sW iU C jcj

iC o n s t r a i n t d e f i n i t i o n s

t h e t o t a l t i m e o f w o r k - b r e a k s b e c a u s e o f d i s c o n t i n u i t i e s o f t h e ta s k ib e t w e e n s u c c e s s iv e u n i t st h e c o m p l e t i o n t i m e o f p r o j e c t u n i t jt h e t i m e u n i t c o s t p e n a l t y o r f i n a n c i a l) f o r c o m p l e t i o n d e l a y o f u n i t j .t h e t i m e u n i t c o s t f o r w o r k - b r e a k o f r e s o u r c e s u s e d i n ta s k i .

T a s k d u r a t i o n c o n s t r a i n t s f j = s o + d oP r o j e c t l i n e a r i t y c o n s t r a i n t s so+l >_ f j

t a s k in u n i t j + l f o l l o w s t h e s a m et a s k i n u n i t j . E x c e p t i o n s t o t h isr u le c a n b e h a n d l e d a c c o r d i n g l y )T a s k d e p e n d e n c i e s so > f k j

t h e e x a c t f o r m o f th e c o n s t r a i n td e p e n d s o n t h e d e p e n d e n c y t yp e .)U n i t c o m p l e t i o n t im e :

U C , d e f i n e s t h e p r o j e c t d u r a ti o n )R e s o u r c e d e l a y f o r t a sk i :

V i - -1 , 2 . .. .. m a n d j = l , 2 . .. .. n 1 )V i = 1 , 2 . . .. m a n d j = l , 2 . .. .. n - 1 2 )

V i = 1 , 2 . .. . m , j = l , 2 . .. . . n a n d k ~ P i 3 )

U C y > ) ~ j V k ~ E 4 )n 1

W B i = Z s~+, - L ) , v i = 1 . . . . m 5 )j=l

T h e f o l l o w i n g s e t o f c o n s tr a i n t s d e s c r i b e s t h e o p e r a t i o n o f a c t iv i t ie s i n a n L R P :


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P. Ipsilandis MultiobjectiveOptimization n L inearRepetitiveProjectScheduling 26mT o t a l r e s o u r c e d e l a y : W B = ~ W B i (6 )

i

l o b a l o b j e c t i v e f u n c t i o nn m

M i n i m i z e Z c i . ( U C j - M i n U C j ) + ~ f .WB (7)j-1 i=l

w h e r e M i n U C j is t he C P M c o m p l e t io n t im e f o r u n i t j .D e p e n d i n g o n t h e v a lu e s o f th e p a r a m e t e rs cj a n d f t h e a b o v e d e f i n e d g l o b al o b j e c ti v ef u n c t i o n c a n b e u s e d a c c o r d i n g l y f o r o p t i m i z i n g v a r i o u s p r o j e c t o b j e c t i v e s a s l i s te d i ntable 1 .

a b l e 1 . Pa ram eter se t t ins fo r var ious ob jec tive func t ionsO b j e c t i v e f u n c ti o n M i n im i z e P a r a m e t e r v a l u e sPro ject DurationTotal work-b reak t imeTotal Cost ofresource work-breaksUnit Completion TimeCos t o f Delays inComple t ionTradeoffs be tweenCost of complet iondelays and resourcework-breaks

U n On=l, rest o f cj and f (7.1)equal 0WB All ieq ua l 1, all cj (7.2)equal to 0All cj equal to 0 (7.3)Z W Bi=l

m Al l feq ua l O, a ll @ (7.4)Z U C i equ al to 1i=1m

2 c i ( g c i - - M i n U C i )i=ln mZci. UCj-MinUCj)+ f . W B ij=l i=l

All equal to 0 (7.5)

(7.6)

4 . A p p l i ca t i o n R esu l t s o f M O L P S L R P4 .1 A c a se s t ud y e x a m p l e o f L R PT h e L P m o d e l th a t w a s d e f i n e d i n t h e p r e v i o u s s e c t i o n i s a pp l ie d t o a c a s e s t u d y o f as m a l l l i n e a r d i s c r e t e p r o j e c t t h a t w a s i n i t i a l l y u s e d b y t h e R S M a u t h o r s [ H a r r i sI o a n n o u ( 1 9 98 ) ], w i t h s o m e m o d i f i c a t i o n s t o t a s k d u r a t i o n ti m e s . T h e p r o j e c t c o n s i st so f s i x r ep e t i t i v e u n i t s , e ach h av i n g s i x d i s c re t e ac t i v i t i e s r ep ea t ed a t e ach u n i t . A l lt a s k d e p e n d e n c i e s a r e f i n is h - t o - st a r t a n d e a c h a c t i v i t y is p e r f o r m e d b y a s p e c i f ic c r e w .F i g u r e 1 sh o w s t h e p r e c e d e n c e n e t w o r k f o r o n e u n it , th e d u r a t io n o f e a c h t a sk a t e a c ho f t h e s ix p ro j ec t u n i t s an d o t h e r p ro j ec t s p ec i f i c co n s t r a i n t s .


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6 Op er ati on al esearch. An International Journal / Vol.6 No.3 / September - December 2006

Du io Lag24-4-6-6-4-4Duration2-2-2-2-X-2

* Duration of activity a t each of the6 projec t unitsActivity C is not executed at the~ h unit

J Duration. ~ ~ 10-10-10-10-10-10

Duration6 6 6 6 6 6

Due to technological constraint activ ity C canno t start until two days aft er activ ity A isfinished Lag = 2)There is a 5 days delay in activity B between units3 and 4

Figure 1: Precedence network in Unit 14.2 M inimizing work break imeThe CPM schedu le fo r the p ro jec t p roduces a min im um dura t ion o f 48 days . G iventhe 48 duration as a constraint UC6=48), the LP model was run se t t ing work-breakminimizat ion (7 .2) as the object ive funct ion. The resul t ing schedule is shown inf igure 2. The m in im um pro jec t dura t ion o f 48 days can be sa t is f ied wi th a min imu mo 26 days o f wo rk break in par ts of act iv i t ies B and C. Fur the r reduct ion of work -break t ime at act iv i t ies cannot be ach ieve d without extendin g the project s durat ionbeyon d 48 days .

Figure 2. Linear Schedule Minimization of Resource Delays with CPM duration


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P Ipsilandis / Multiobjective Optimization in Linear Repetitive Project Scheduling 63

I f the CPM durat ion cons t raint is re laxed the w ork-breaks can be fur ther reduced to am ini m um of 5 days but th is wi l l resul t in extendin g the project s d urat ion to 62 daysas sho wn in f igure 5 . The w ork break o f task C is e liminated, whi le that of task B isredu ced to 5 days as set by the task s techn ologica l constraints. The f inishin g t ime ofall units is also pulled to 14 days later than in the p revious sched ule.

Figure 3 Minimization of Resource Delays with relaxation of CPM duration4 3 M in im izin g un i t complet ion t imeUnit completion t ime is also an important cr i ter ion in scheduling repeti t ive projects .Co mp letin g units as early as possible could be cri tical because of seasonal co nstraintsand could also af fect the f inancial s tanding of the projects in cases where paym entsdepend on the del ivery of completed uni ts . In l inear projects l ike highwaycons t ruct ion the ear ly del ivery of cer ta in uni ts could m ark the p roject s incomegenerat ion, thus any delay in uni t complet ion could cha nge negat ively the net presentvalue of the project . In this case the objective is set to minimization of completiont ime of a l l or cer ta in uni ts , even i f that means that addi t ional work-brakes must beinser ted at tasks . Figure 4 shows the resul t ing schedule when minimizat ion ofcompletion t ime for al l units (7.4) is defined as the objective of the MOLPS / LRP.Eve n though the overal l durat ion of the project cannot be fur ther reduc ed to less than48 days, intermediate units are delivered at earl ier t imes. (The nu mb ers in parenthesisshow the gains in del ivery t ime for each uni t as compared to the RSM schedule infigure 4) . On the other hand, these savings are achieved by sacrif icing workcontinuity. The total work-break t ime in al l act ivit ies is increased to 64 days, or 38days m ore than m ini m um work break t ime that can be achieved for the same projectduratio n (f igure 4).


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64 Opera t iona l Resea rch . An In te rna tiona l Journa l / Vol .6 No.3 / September - Decem ber 2006

Figure 4. Linear Schedule Minimization of Unit Completion Time4 .4 Cost r ade o f f s between wo r k b r a ke and un i t du r a t i o nIn real l ife projects the cost of wor k-bre aks at different activities varies acco rding tothe type and scarceness o f the resources consu med. In addi tion, the la teness del iverycos t which af fects the overall cos t of the project in d if ferent ways based o n the overal lschedule of project income receipts and payments could be associated with thedel ivery of the ent i re project and/or with del iver ies of completed project uni ts . Thecost based objec tive functions l is ted in the previous sections [ 7.3), 7,4), 7,6)] canbe used for provid ing opt imal schedules in th is case.Trade -off analys is between co mplet io n t ime and work-break t ime is poss ible wh enassociated cost data exist . However, even in the case that no exact cost estimatesexis t , the tools of LP sens i t iv i ty analys is can be used to provide opt imum trade offranges of the rat io betw een delay and wo rk-break uni t cost . In i t ia lly the cos t of delaysis set equal to the unit cost of the work-bre ak. Since i t is only the relatio nship betw eenthe two cos t parameters and not their absolute values that af fect the opt imizat ionprocess both values are init ially set equal to one and using LP range analysis theopt imal i ty ranges and the changes in the opt imum solut ion can be der ived. Theexamples that fo l low demonstrate th is type of analys is :Tr ade of f between pr oj ect durat ion delay cost and wor k break costThe f i rs t one is a t radeoff analys is between delays in project complet ion and task


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P Ipsilandis / Multiobjective Optimization in Li near Repetitive Project Scheduling 65

project as i t i s se t by the CPM. I t i s a lso assumed that in termediate delays incomple t ing each un i t do no t impose any cos t to the p ro jec t , and tha t the cos t o f work-brea ks is the sam e for all tasks.In th is case the object ive t rad e-off funct ion (7 .6) of the LP m odel c an be wri t ten as :Minimize UC, WBThe range analys is on the object ive funct ion cos t parameters produced the resul tsshow n in f igure 5 . W he n the work -break uni t cos t is below 50 of the cos t paid forp ro jec t de lays the op t imum schedu l ing p roduces a p ro jec t dura t ion o f 48 days(min im um poss ib le ) wi th a max imu m work-break t ime o f 26 days . Whe n the work-break uni t cos t is bet we en 50 and 100 o f the project delay cos t , i t i s mo reecon om ical to le t the project s l ip by 5 days in order to reduce work -breaks to 16 days .And f ina l ly when the work-break cos t exceeds the p ro jec t de lay cos t , the op t imumschedu le i s the one tha t in t roduces the min im um of 5 days o f work-breaks , whichresul ts in extending the project durat ion by 16 days .

Fig ure 5. Tradeoffs between proje ct comp letion and work-breaks.Tr ade o f f between un i t du ra t i on delay cost an d wor k b r ea k costIn the second example, i t i s assumed that a penal ty cos t is associated with delays inde l ivery t ime o f ind iv idua l p ro jec t . De lays cou ld be measured as dev ia t ions f rom theear l ies t f in ish dates o f the project uni ts or f rom an ag reed t ime. This choic e wil l notaf fect a t a l l the range analys is that fo l low, s ince the cos t coeff ic ients of the object ivefunc t ion remain unchanged . For s impl ic i ty purposes here we as sume tha t the samepen al ty c appl ies to dela ys in any uni t. Also as in the previous case , the cos t of work-breaks is assumed to be the same for a ll tasks.


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66 Opera t iona l Resea rch . An In te rnat iona l Journa l / Vol .6 No.3 / Septem ber - December 2006

In this case the objective fun ction 7.6) of the LP mod el is set to:M inim ize Z c . U C j - D y ) + Z T . W B i = c m o i - c D j

j=l i=1 ~j =l = j=lThe range analys is results are sh own in f igure 6. Minim izing the work brake t ime inthe task is necessary only whe n the associated work-break cos t i s 6 t imes higher thanthe cost paid for u nit delays 4.A s ignif icant break point i s whe n work-brea k cos t i s 3 t imes h igher than cos t ofdelays. Un der this level total delays in the units are kept below 17 days in total about3 da ys per unit) wh ile above this level , they range from 51 to 114 days about 8,5 to19 days) . Figure 5 gives a graphical representation o f the results .

Figure 6. Tradeoffs between unit completion delays and work-breaks.5 Conc l u s i o n sSchedul ing of l inear repet it ive cons t ruct ion projects i s not a s ingle dim ens ion decis ionprocess . Factors such as the durat ion of the project the del ivery o f the individualproject uni ts on t ime, the cont inuat ion of resource usage, and the s lack t ime arefactors that cons t ruct ion managers must take into cons iderat ion in deciding anopt imum schedul ing for the project . A schedul ing decis ion must take intocons iderat ion more than a s ingle factor and most o f the t imes t radeoffs are requiredbetween uni t co mp let ion t imes , project durat ion and work-breaks. St ric tly schedu l ingmethods like CPM and RS M can prov ide main ly answers r egard ing t ime d imens ionbut cannot address in an integrated way, cost considerations that are raised s incemany cos t factors such di f ferent cos t of work-breaks a t each task, penal ty cos tsrelated to delays in project duration, f inancial costs ar is ing from inadequ ate cash f low


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P Ips il and is / M u l t iob jec t ive Op t imizat ion in Linea r Repe t i t ive Pro jec t Sche du l ing 67

because o f la t eness in d e l i ve ry of pa rt ia l un it s. The M OLP S-LRP m ode l can addre ssthese i s sues and prov ide answers t o re la t ive ques t ions . Fur the rmore t he M OL PS-LR Pc a n be u s e d t o de t e r m i ne t he op t i m um nu m be r o f s e gm e n t s i n a l i ne a r p r o j ec t w he nsegm ents a re not def ined as phy s ica l uni t s ( i.e . f loors , apar tments e tc . ) g ive n tha t aninc rease i n t he n um ber o f un i ts a f fect s pos i t ive ly the d ura ti on of t he p ro j ec t i n a wayo f d imin i sh ing re tum s b u t a l so i nc rease s the cos t o f w ork-breaks and t he to t a le m p l oym e n t o f t he r es ou rc e s.eferences

A G u i de to t he P r o je c t Ma na ge m e n t B ody o f K now l e dge , 3rd Ed i t ion, (2004). PM IStandard Co m m it tee , PM I Insti tu te .A s h l e y , D B (1980) , S imu la ti on of r epe t i ti ve -un i t cons t ruc t ion Journa l o fCon st ruc t ion Divis ion, Vol . 106 , No. CO2, p p. 185-194.Bi rre l l, G. S . (1980) , Co nst ruc t ion plan ning -bey on d the c r it ica l pa th Journa l o fCo nst ruc t ion Div is ion, Vol . 106, No. CO 3, pp. 389-407.C a r l R .I . and M eye r , W.L . (1974), P l ann ing cons t ruc t ion of repe t i t ive bu i l d inguni t s Journa l of Const ruc t ion Divis ion, V ol . 100 , N o. CO3, pp. 403-41 2.Da wo od, N. (1998) , Es t im at ing projec t and ac t ivi ty dura tion: a r isk m ana gem entapproach using ne tw ork ana lys is Cons t ruc t ion Mana gem ent and Econom ics , 16 ,pp. 41-48.Dress ie r , J . (1974) , Co nst ruc t ion schedul ing o f l inear cons t ruc t ion s i t es Journa l o fCo nst ruc t ion D ivis ion, Vol . 100 , No . CO 4, pp. 571-587.Harm e l ink , D . J. and Row ings , J .E . (1998) , L inea r Schedul ing M ode l : deve lopm entof con t ro l li ng ac t iv i t y pa th Journa l o f Cons t ruc t ion Eng inee r ing andM anag em ent , Vol . 124, No . 4 , pp. 263-268.Harm e l ink , D . J. (2001) , L inea r Schedul ing M ode l : fl oa t character is t ics Journa l o fCons t ruc ti on Enginee r ing and M anagem ent , Vol . 127 , No. 4 , pp . 255-260 .Harri s, R .B. and Ioann ou, P .G. (1998) , Sc hed ul ing projec t w i th repea t ing ac t ivi ti esJoum a l o f Cons t ruc t ion Enginee r ing and M anagem ent , Vol . 124 , No. 4 , pp . 269-278.

Ioann ou, P .G. and Harr i s , R .B. (2003), D iscuss ion of a lgor i thm for de te rm iningcont ro l li ng pa th cons ide r ing re source con t inu i ty Journa l o f Co m put ing i n C iv i lEn gin eerin g, Vo l . 17 ,No. 1, pp. 68-70.Ioanno u, P .G. and Y ang , I .T . (2003), D iscuss io n o f a lgor i thm for de te rm iningcont ro l li ng pa th cons ide r ing re source con t inu i t y Joum a l o f Com put ing i n C iv i lEn gin eerin g, Vo l . 17 ,No. 1, pp. 70-72.Ioannou , P .G. and Y ang , I .T . (2004) , Di scuss ion of com par i son of L inea rS c he du l ing M ode l a nd R e pe t i ti ve S c he du l i ng M e t hod J oum a l o f C ons tr uc ti onEng inee r ing and M anagem ent , Vol . 130 , No . 3 , pp . 461-463 .Johns ton , D .W. (1981) , L inea r Schedu l ing M e thod for h ighw ay cons t ruc ti onJou m al of Const ruc t ion Divis ion, Vo l . 1 07, No . CO 2, pp. 247-261.


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Ya ng , I.T . a nd I oa nno u , P .G . (2001) , R e sou r c e - d r ive n sc he du l ing f o r r e pe t i ti vep r o je c ts : a pu l l - sy s t e m a pp r oa c h , P r oc e e d ings o f t he 9 th I n t e r na t iona l Gr oup f o rLe a n C ons t r uc t ion C onf e r e nc e , S inga por e , 6 - 8Augus t , pp . 365 - 377 .Ya ng , I .T . a nd I oa nnou , P .G . ( 2004) , S c h e du l ing w i th f oc us on p r a c t i c a l c onc e r ns inr e p e ti ti v e p r o j e c ts C o n s t r u c t i o n M a n a g e m e n t a n d E c o n o m i c s , V o l . 2 2 , p p . 6 1 9 -630.Ya ng , I.T . ( 2002a ), R e p e t i t i ve P r o j e c t P l a nne r : r e sou r c e - d r ive n sc he du l ing f o rr e p e ti ti v e co n s t r u c ti o n p r o j e c ts P h D . D i s s er t at i o n , U n i v e r s i t y o f M i c h i g a n , A n nA r b o r , M I , U S A .Ya n g , I.T . (2002b) S toc h a s t i c a na ly s i s on p r o j e c t du r a t ion unde r t he r e qu i r e m e n t o fc on t inuous r e sou r c e u t i l iz a t ion , P r oc e e d ing s o f the 10 th I n t e r na t iona l Gr oup f o rL e a n C o n s t ru c t io n C o n f e r e n c e , G r a m a d o , B r a z i l , A u g u s t , p p . 5 2 7 - 54 0 .


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multiobjective optimization in linear repetitive project scheduling

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