A robust optimisation approach to project scheduling and resource allocation. Elodie Adida* and Pradnya Joshi

Transcription

1 In. J. Servces Operaons and Informacs, Vol. 4, No. 2, A robus opmsaon approach o projec schedulng and resource allocaon Elode Adda* and Pradnya Josh Deparmen of Mechancal and Indusral Engneerng, Unversy of Illnos a Chcago, Chcago, IL, USA Emal: [email protected] Emal: [email protected] *Correspondng auhor Absrac: In a rapdly changng and hghly unceran envronmen, Per ne-based projec managemen sysems can be used n he reschedulng (conrol reconfguraon) of projecs, when unforeseen changes occur or new daa esmaes become avalable. However, havng o reschedule he projec afer has sared can resul n sgnfcan subopmaly due o ncreased cos. In hs paper, we presen a soluon mehod whch s, o a large exen, robus o daa uncerany and does no requre reschedulng for a ceran predefned level of uncerany. We buld he model usng robus opmsaon echnques whch address daa uncerany n projec parameers, by akng advanage of rsk poolng and whou knowledge of her probably dsrbuons. We llusrae ha he oal cos when he robus soluon s used s generally lower han he cos of reconfgurng he deermnsc soluon, or han a penaly cos due o overme of he deermnsc soluon, for a hgh enough penaly cos per me un. We fnd ha he robus soluon s beer proeced agans consran volaons, ncludng me and cos overruns. Keywords: opmsaon under uncerany; robus opmsaon; resource allocaon; projec managemen; Per nes; conrol reconfguraon. Reference o hs paper should be made as follows: Adda, E. and Josh, P. (2009) A robus opmsaon approach o projec schedulng and resource allocaon, In. J. Servces Operaons and Informacs, Vol. 4, No. 2, pp Bographcal noes: Elode Adda s an Asssan Professor n he Deparmen of Mechancal and Indusral Engneerng a he Unversy of Illnos a Chcago. She receved her PhD n Operaons Research from he Massachuses Insue of Technology n 2006, and a Dplôme d Ingéneur from Ecole Cenrale Pars, France, n Her research neress nclude mahemacal programmng, nonlnear dynamc opmsaon, game heory, opmsaon under uncerany and applcaons of operaons research o supply chan and healh care. Pradnya Josh receved her Maser of Scence Degree n Indusral Engneerng from he Unversy of Illnos a Chcago n She holds a Bachelor of Engneerng n Mechancal Engneerng from Osmana Unversy, Inda. Copyrgh 2009 Inderscence Enerprses Ld.

2 170 E. Adda and P. Josh 1 Movaon, leraure revew and conrbuon 1.1 Movaon Modellng daa uncerany s an mporan aspec of Operaons Research. In projec managemen sysems, he resuls ha are obaned assumng ha he projec parameers are deermnsc are no longer vald when hese parameers ake realsed values dfferen han expeced. Daa may lack precson, renderng he compued opmal soluon subopmal, or even nfeasble by leadng o consran volaons. I s herefore mporan o ensure ha he plannng and resource allocaon ools ake hs uncerany no accoun and allow schedulng and resource allocaon ha s robus o daa uncerany, so ha he schedule remans feasble and close o opmum even f daa s changed slghly. For example, n he afermah of a dsaser, decson-makers somemes have o make decsons based on somewha naccurae and consanly updaed daa. The effcency of he rescue acons underaken and he adequae allocaon of he scarce resources avalable, grealy affec he number of casuales. I s crucal o develop approprae plannng ools so as o be prepared o respond n he mos effcen fashon. The schedulng and assgnng of resources n projec managemen sysems s done effecvely usng Per ne-based supervsory conrol reconfguraon echnques (Haj and Darab, 2007). A Per ne supervsory conroller s developed ha enforces he ask precedence relaonshps and he resource consrans for a projec. In he even of any change n he acual progress from he planned progress, he developed supervsory conroller s reconfgured and he new soluon s used o conrol he projec. Ths allows fas reschedulng n response o new nformaon, bu may affec he cos of he projec grealy. Our goal s o compue a soluon ha s close o opmal even as he daa s perurbed, hus elmnang he need for reconfguraon o a ceran exen. Sochasc opmsaon s he mos radonal way o address daa uncerany (Danzg, 1955). However, deermnng he probably dsrbuon of he unceran parameers s no always easy and may be somemes unrealsc. Robus opmsaon has been nroduced n he las decade as an alernave o sochasc opmsaon (Bersmas e al., 2004). The movaon behnd robus opmsaon s, on he one hand, o no requre a parcular probably dsrbuon on he daa and on he oher hand, o fnd an alernave o overly conservave wors-case reasonng, n whch one assumes ha he value of he daa ha s he leas favourable occurs for all unceran parameers a all mes. Inroducng a budge of uncerany on he daa s an effcen way o measure he rade-off beween conservaveness and performance (Bersmas and Sm, 2004). The budge of uncerany represens he overall cumulave amoun of varaon away from nomnal values ha mus be shared among unceran parameers. The budge of uncerany s npu n he model; he modeller can decde wheher he/she wans o oban a more conservave soluon (by choosng a large budge of uncerany) whle sacrfcng opmaly, or a soluon ha performs well and s less mmune o daa uncerany (by choosng a smaller budge of uncerany). The movaon behnd such a model s o ake advanage of rsk poolng and proecng he sysem only agans he mos lkely oucomes. 1.2 Leraure revew The Operaons Research leraure reas he presence of daa uncerany n opmsaon problems n several ways. The problem s somemes solved assumng all parameers are

3 Projec schedulng and resource allocaon 171 deermnsc; subsequenly sensvy analyss s performed o sudy he sably of he nomnal soluon wh respec o small perurbaons of he daa. Sochasc programmng s used when he probably dsrbuon of he underlyng unceran parameers s known. In he las decade, some researchers have focused on reang uncerany n an alernave way, known as robus opmsaon. A robus opmsaon formulaon was frs consdered by Soyser (1973) n he case of a lnear opmsaon problem where he daa were unceran whn a convex se. He addresses uncerany by akng a wors-case approach and was crcsed of beng overly conservave. Ben-Tal and Nemrovsk (1998, 1999, 2000) address he ssue of over-conservaveness by consderng daa uncerany ses ha are ellpsods. Bersmas and Sm (2004) nroduce he noon of budge of uncerany o conrol he level of conservaveness. Bersmas e al. (2004) use uncerany ses descrbed by an arbrary norm. Bersmas and Brown (2005) consruc uncerany ses for LPs by akng a coheren rsk measure on prmve. Bersmas and Sm (2006) propose a relaxed robus counerpar for general conc opmsaon problems. The robus opmsaon mehodology has been appled o a number of areas. Bersmas and Sm (2003) use he robus opmsaon approach for dscree opmsaon and nework flow problems. Bersmas and Thele (2006) apply robus opmsaon prncples o nvenory heory and supply chan managemen. Ben-Tal and Nemrovsk (1997) use robus opmsaon for russ opology desgn. Goldfarb and Iyengar (2003) apply robus opmsaon o porfolo selecon problems. Ben-Tal e al. (2005) ake a robus opmsaon approach for mul-perod sochasc operaons managemen problems, and n parcular he realer suppler flexble commmen problem wh unceran demand. Adda and Peraks (2006, 2009) apply he robus opmsaon mehodology o a problem of dynamc prcng and nvenory conrol. They nvesgae he dfferences wh a sochasc opmsaon formulaon of he problem n Adda and Peraks (2008). Over he las few decades, researchers have developed several projec managemen ools ha can be used for resource allocaon and schedulng of projecs (Ayug e al., 2005; Deblaere e al., 2006). Projec schedulng and projec managemen problems are ofen solved usng nework dagrams such as he crcal pah mehod (Angus e al., 2000). Per nes represen a good alernave o manage projecs where schedules depend on resource allocaon and have o be replanned over me (Per, 1966). Muraa (1989) descrbes he hsory, properes, analyss mehods and applcaon areas of Per nes. A varey of papers descrbe how Per nes can be used o model he projec dynamcs (see e.g. Mago, 1989; Km e al., 1995; Kumar and Ganesh; 1998; Jeeendra e al., 2000; Reddy e al., 2001; Sampah, 2004; Gullo and Agoson, 2007; Haj and Darab, 2007). Kumar and Ganesh (1998) llusrae ha Per nes provde a powerful formalsm for represenng and analysng concurren sysems. They use Per nes o faclae resource allocaon n projecs under some condons commonly encounered n pracce. Reddy e al. (2001) address he use of Per nes as a schedulng and modellng ool n mul-mode and mul-resource consraned projec schedulng. In Km e al. (1995), Per nes are used o model consrucon projec asks bu no for projec monorng. Sampah (2004) dscusses Per ne-based supervsory conrol reconfguraon echnques. Haj and Darab (2007) show he applcaons of Per ne-based supervsory conrol reconfguraon echnques n he conrol of projecs when he projec progress s known. They develop a supervsory conroller whch can be reconfgured o modfy and revse he plan based on he acual progress.

4 172 E. Adda and P. Josh 1.3 Conrbuon 1 We consder he deermnsc, he reconfguraon and he robus opmsaon approach. We sudy he advanages of each approach and show he dfferences n he soluons hey provde. 2 We provde an equvalen deermnsc formulaon o he robus model: he robus counerpar problem. 3 The model ncorporaes uncerany no only n he cos coeffcens bu also n he duraon of each ask alernave. 4 We compare n a numercal sudy he complexy of he dfferen approaches and her performance n erms of consran volaons and realsed cos. In Secon 2, we nroduce he orgnal deermnsc model used n he conrol and reconfguraon of a projec managemen sysem. In Secon 3, we nroduce he robus formulaon and develop he robus counerpar problem. In Secon 4, we carry ou a numercal sudy o compare he wo models on a specfc numercal example and nerpre he resuls. 2 The model In hs secon, we nroduce he Per ne-based supervsory conroller and gve a bref descrpon of he neger program used for he conrol and reconfguraon of he supervsory conroller. Ths model was developed by Haj and Darab (2007). The deals can be found n Gullo and Agoson (2007) and Haj and Darab (2007). 2.1 The deermnsc model We consder a projec nvolvng n asks a 1, a 2,, a n (Haj and Darab, 2007). Each ask uses some or all of m dfferen ypes of resources for s execuon. R k represens he avalable number of resource ype K. Based on he amoun of resource ulsed, each ask realses a ceran execuon speed, ou of he dfferen duraon alernaves. Dfferen coss are assocaed wh he dfferen alernaves. For example, consder a medcal suppor sysem wheren a paen mus undergo ceran medcal asks before he leaves he sysem (Fgure 1). The workflow shows all he asks ha mus be performed before he paen s dscharged. I also gves he precedence relaonshps of hese asks. There are wo ypes of resources n hs sysem: docors and nurses. All asks do no use boh he resources. For example, he ask nurse vs does no need any docors. Each ask may be performed n dfferen duraons of me. The number of nurses and docors requred s dfferen for dfferen duraon alernaves and hence dfferen alernaves ncur dfferen coss. If he number of docors and nurses avalable s no suffcen, addonal resources can be hred a a ceran cos. The schedulng of each ask, he duraon of each ask and number of addonal resources o hre may be opmsed under some consrans on he sysem lke an assgned fnsh me for he job.

5 Projec schedulng and resource allocaon 173 Fgure 1 Medcal suppor sysem example The modellng of such a workflow s done usng Per nes. A Per ne s defned by places, ransons and dreced arcs. The npu o a ranson s a place from whch an arc runs o a ranson and an oupu of a ranson s a place o whch an arc runs from a ranson. Places conan okens, whch deermne f a ranson can be fred, or no. Every arc has he wegh assocaed wh. A ranson s enabled or fred f he number of okens n an npu place s greaer han he wegh of he arc connecng he npu place o he ranson. The dsrbuon of okens over a place s called a markng. In Muraa (1989), he chemcal reacon producng waer s llusraed. The ranson s enabled as wo okens n each npu place, showng wo uns of H 2 and O 2, are avalable. Afer frng, he markng changes such ha he ranson can no longer be fred (Muraa, 1989) (See Fgure 2). Fgure 2 An llusraon of ranson frng: (a) markng before ranson s enabled and (b) markng afer frng ranson, s dsabled (Muraa, 1989) I s he npu sae-ranson marx and represens he amoun of okens aken by each ranson from s npu places when he ranson s fred (Haj and Darab, 2007). O s he oupu ranson marx and represens he amoun of okens sen by each ranson o s oupu places when he ranson s fred. Incdence marx of a Per ne s defned by D P P = O I. The sae equaon D σ = M M0 nvolves he nal markng of he Per ne (M 0 ), he arge markng o be reached (M) and he number of mes each ranson s be fred o reach he arge (frng coun vecor σ). The Per ne showng he behavour of he unconrolled sysem s called a plan Per ne. The conrol specfcaons are mposed on he plan Per ne as a se of lnear consrans on he markng of he plan Per ne places. The supervsory conroller, used o resrc he behavour of he plan Per ne so as o sasfy he consrans, s consruced by addng approprae number of

6 174 E. Adda and P. Josh conrol places o he plan Per ne and lnkng hese places o he requred exsng ransons. Durng runme, he coeffcens n he consrans may change causng a change n he conrol specfcaons and a reconfguraon s requred. To mplemen he reconfguraon, for every conrol ranson, a unque ranson, called he conrol ranson, s defned. If he curren sae of he plan s nfeasble, represened by negave markngs n he conrol places based on he new specfcaons, we fre he new conrols (or plan ransons) o urn he negave markngs posve. There s a cos assocaed wh frng he ransons. The objecve of reconfguraon s o remove he negave okens by execung he mnmum cos frng coun vecor of ransons. Ths vecor s found by solvng an Ineger Programmng (IP) model. The followng IP model s used n conrol and reconfguraon of dscree even sysems (Haj and Darab, 2007). The model s an neger program correspondng o he sysem whou explc me dmenson. Ths s done by assocang wo mes wh each ranson, defnng he sac nerval. The frs me value denoes he mnmal me elapsed from enablng he ranson o frng ; he second denoes he maxmum me he ranson can wa before fres. Thus one has o esablsh wo me values for each ranson. The model s racable and s small sze allows o be solved n a shor perod of me. Mnmse cσ (1) H Subjec o p p p D 0 Y M σ c = c c D I Y M (2) p σ 0 for, σ T p (3) p S F 0 for a, a A (4) j j 2, n σ s a 0 for every j= 1 F S d a (5) F F 0 a (6) σ end= 1 where end p erm a (7) F S 1 + M( x 1) (8) j F Sj Mx (9) F S 1 + M( x 1) (10) j j Fj S Mxj (11) ( x + x + x + x + x + x ) 6 + M( x 1) (12) j k k jk kj jk ( x + x + x + x + x + x ) 5+ Mx (13) j k k jk kj k

7 Projec schedulng and resource allocaon 175 n ( x ) n M ( xn 1) 1 n 1 + = (14) n ( xn 1) n 1+ Mxn (15) = 1 n n p r pk σ p R + k σ + k M(2 x xj ); n = 2; k = 1,2,3..., m; P (16) p = 1, 2,3..., n and n= number of overlaps p rpk σp Rk + σk ; k = 1,2,..., m (17) p n n p ( rpk σp ) xn Rk + σk ; k = 1,2,, m (18) p σ = 0,1 for plan ransons (19) σ 0 and neger for conrol ransons (20) p c Y, Y, S, F 0 and neger (21) x, x j, x n bnary and M a large neger posve number In he above model, he objecve (1) s o mnmse he coss of frng ransons (c ) n he Per ne. The varable σ represens he elemens of he frng coun vecor and shows f he ranson s seleced for frng. The oal cos s obaned over all H, where H s he se of all plan ransons T p and all conrol ransons T c. Consran (2) ses he curren markng of he Per ne o a arge markng and s a represenaon of he sae equaon. D p s he ncdence marx of he plan Per ne, D c he conroller ncdence marx, Y p he arge markng of he plan places, Y c he arge markng of he conrol places, M p he curren markng of he plan Per ne and M c s he curren markng of he conrol Per ne. Consran (3) guaranees ha a ask s execued only f enough of he resources requred are avalable,.e. a ranson s fred only f all of s npu places have enough okens. Here, * s he se of npu places of ranson and * p s he se of npu ransons of place p. Consran (4) ensures ha a ask canno begn unl all he prerequse asks have been execued. A 2, s he se of all acves ha a s a drec prerequse for. S j s he sar me of acvy j and F s he fnsh me of acvy. Consran (5) forces he end me of a ask o be no less han s duraon added o s sarng me. d s he duraon of acvy wh resource se up j and n s he number of resource-duraon alernaves. Consran (6) enforces a gven deadlne (F) on he fnsh me of he projec. Consran (7) ensures ha he projec s execued,.e. no execung he asks s nfeasble. As hs s an neger program whou me dmenson, s soluon may correspond o a resource allocaon ha s no feasble: due o he overlappng of asks, mulple acves smulaneously use a parcular knd of resource, requrng more resource han s avalable. Therefore, addonal resources need o be hred for projec

8 176 E. Adda and P. Josh compleon. Overlap consrans are nroduced o know f he asks are overlappng and based on ha, o fnd ou he maxmum amoun of resources he projec needs a a gven pon. Consrans (8) (11) represen he overlap resrcon for wo acves. For he wo asks, j ha may be overlappng, wo dfferen varables x and x j are nroduced. There s an overlap beween wo asks f he value of boh varables s equal o one. Consrans (12) and (13) show he overlap resrcon for a se of hree acves. Consrans (14) and (15) generalse o an overlap of n acves. Consran (16) forces he model o hre exra resources f here are no enough avalable when wo asks overlap. Here, r pk s he amoun of resource k needed for he acvy usng he ranson p of he plan Per ne, σ p he elemen of he frng vecor for a ranson p, σk he conrol ranson of he kh resource and R k s he markng of he conrol place whch gves he avalably of resources. Consrans (17) and (18) generalse o an overlap of n asks. Consran (19) gves bnary condon for plan ransons and neger condons for he number of resources o hre or fre. The deals of he model can be found n Gullo and Agoson (2007) and Haj and Darab (2007). 3 The robus formulaon In compung a soluon o he above model, he daa s assumed o be precsely known. Bu he cos coeffcens c and he duraon coeffcens d are subjec o uncerany. Due o uncerany n he cos parameers, he oal cos of he projec may be hgher han wha would have been expeced n he deermnsc model. Uncerany n he duraon coeffcens may resul n fnsh me consran volaons for each ask (consran 5) and volaon of projec deadlne consran (consran 6). In some applcaons, volang he projec fnsh me may be permed bu would ncur a penaly per me un. In oher applcaons, f he fnsh me mus be me a reconfguraon s necessary. The cos may be ncreased sgnfcanly n eher of hese wo cases. Snce we have a varey of numercal parameers subjec o uncerany (ask duraons and ask coss), we nroduce a budge of uncerany for each caegory: a ask duraon budge for each ask (o be shared by he mulple resource-duraon alernaves for ha ask) and a ask cos budge (o be shared by all he cos coeffcens). The use of rsk poolng n hs model llusraes ha for each ask s unlkely ha all duraon alernaves esmaes are wrong. However, ask duraons do no share he budge wh duraons of anoher ask or wh cos parameers, as hese unceranes come from radcally dfferen sources and do no necessarly balance each oher ou. We derve a robus model by consderng he followng uncerany framework (Bersmas and Sm, 2003, 2004). We assume ha he realsed value of he cos coeffcen c akes values n [ c, ˆ c + c] (where c s he nomnal value and c ˆ s he lengh of he range of uncerany) and ha every realsed value of he duraon coeffcen d akes values n [ d dˆ, d + dˆ ] (where d s he nomnal value and j d ˆ s he half-lengh of he range of uncerany). Le J be he se of ndces of duraon coeffcens d ha are subjec o uncerany.e. J { ˆ j d 0} = >. Smlarly, J 0 represens he se of ndces of duraon coeffcens c ha are subjec o uncerany,.e.

9 Projec schedulng and resource allocaon 177 { 0} J = c >. We nroduce budges of uncerany Γ for he duraon coeffcens of duraon alernaves of ask and Γ 0 for he cos coeffcens, where Γ [0, J ] and Γ 0 [0, J 0 ]. The budge of uncerany for ask means ha a mos Γ of he coeffcens are allowed o realse whn her enre range and one-coeffcen d s allowed o devae by a mos ( Γ Γ ) ˆ d (Bersmas and Sm, 2003, 2004). These range of realsaon consrans and budge of uncerany consrans defne he uncerany se of he unceran daa. Our goal s o fnd a soluon ha s close o opmum and feasble for any realsaon of he daa whn hs se. 0 ˆ Le Δ be he se of all realsaons { d, = 1,2,, n, j J} sasfyng he followng consrans: n he uncerany se,.e. d [ ˆ, ˆ d d d + d ] d d Γ ˆ j J d We wan consran (5) o be sasfed for any realsaon of he daa whn he uncerany se Δ. Thus he robus formulaon of consran (5) s gven by: { } F S d σ s a 0, a,, 1,..,, d = n j J Δ (22) j J Ths represens an nfne number of consrans, as he nequaly has o be sasfed for any realsaon of he duraon parameers whn he uncerany se, whch ncludes an nfne number of pons. We oban a deermnsc counerpar of hs consran by deermnng he realsaon ha represens he wors case (whn he uncerany se) for sasfyng hs nequaly,.e. by solvng a subproblem ha searches for he mnmum of he lef-hand sde above: F S d σ mn { d } Δ s a j J Noce ha hs wors realsaon depends on he choce of decson varable σ of he orgnal problem. Ths ransformed consran s hen plugged back no he orgnal problem n he place of consran (22). Ths manpulaon ensures ha he orgnal consran nvolvng unceran duraons wll be feasble for any realsaon whn he uncerany se defned by he range of uncerany consrans and he budge of uncerany consran. We proceed n a smlar way wh unceranes n he cos coeffcens. We oban ha he robus counerpar of he Per ne neger model s: (23) Mnmse cσ + z Γ + q H J0 Subjec o p p p D 0 Y M σ c = c c D I Y M (24) p σ 0, for, σ T p (25) p

10 178 E. Adda and P. Josh S F 0, for a, a A (26) j j 2, n σ s a, for every Γ + j= 1 j J (27) F S d z q a z + q cy ˆ, J (28) z + q dˆ y,, j J (29) j y y, j (30) j σ s a j F F 0, a (31) σ = 1 where end p erm a (32) end * F S 1 + M( x 1) (33) j F Sj Mx (34) F S 1 + M( x 1) (35) j j Fj S Mxj (36) ( x + x + x + x + x + x ) 6 + M( x 1) (37) j k k jk kj jk ( x + x + x + x + x + x ) 5+ Mx (38) j k k jk kj k n ( xn 1) n+ M( xn 1) (39) = 1 n ( xn 1) n 1+ Mxn (40) = 1 n n p P r σ R + σ + M(2 x x ); n= 2; k = 1,2,3,, m; pk p k k j p = 1, 2, 3,, n and n= number of overlaps p (41) n p rpk σp Rk + σk, k = 1,2,, m (42) p n n p ( rpk σp ) xn Rk + σk, k = 1,2,, m (43) p σ = 0,1 for plan ransons (44) σ 0 and neger for conrol ransons (45) p c Y, Y, S, F 0 and neger (46) y, z, q, q 0, j J, J (47) j 0 0 x, x j, x n bnary and M a large neger posve number

11 Projec schedulng and resource allocaon 179 The robus counerpar s smlar o he orgnal problem n he sense ha s sll a lnear program. Bu he followng dfferences can be observed: The number of consrans and he number of varables has ncreased. We nroduce new varables y j, z, q, q 0 and z 0. There s an addonal lnear erm n he objecve funcon (23) and he ask fnsh me consran s made gher (27). The addonal erm zγ+ j J q j n he duraon consran ensures proecon by provdng a margn, o allow for changes n he duraon coeffcens so ha when he nex ask s scheduled o begn, he prevous one s more lkely o have ended and hus no reconfguraon s necessary. Smlarly, he lnear erm n he objecve proecs agans changes n he cos coeffcens. However, he overall srucure s smlar and he complexy of solvng he problem has no ncreased. Therefore, he runnng me does no sgnfcanly ncrease when adapng he deermnsc model o a robus model ha proecs he soluon agans daa uncerany. 4 Numercal sudy In hs secon, we descrbe a numercal example (Gullo and Agoson, 2007; Haj and Darab, 2007) o compare he deermnsc model whou reconfguraon, he deermnsc model wh reconfguraon and he robus model. We llusrae he followng: 1 The robus soluon provdes beer proecon agans consran volaons when compared o he deermnsc soluon whou reconfguraon, when daa s perurbed. I s herefore more lkely o allow o fnsh he projec on me. 2 Ths gan n proecon s acheved a a small ncrease n cos. 3 The consran volaon n he deermnsc model may ncur cos ncrease larger han cos ncrease due o robus f a penaly s ncurred for each me un of overme. 4 The robus soluon yelds lower cos han he deermnsc wh reconfguraon f he projec fnsh me mus be me and he deermnsc soluon mus be reconfgured as soon as wll no mee he fnsh me accordng o he curren schedule. 4.1 Example We consder he Per Ne example nvolvng sx acves a 1, a 2, a 3, a 4, a 5 and a 6. The workflow uses wo ypes of resources N and D. The nework dagram of he projec wh s resource-duraon alernaves s shown n Fgure 3 (Gullo and Agoson, 2007; Haj and Darab, 2007). In hs example: 1 The number of resources N, D avalable when he projec sars s 2 each: The resource avalably, ha s he markng of he conrol places, s R k = 2. 2 The projec fnsh me s 8: F = 8.

12 180 E. Adda and P. Josh 3 The cos of hrng resource N (frng he assocaed conrol ranson) s 100 and ha of hrng resource D s The cos of choosng one duraon alernave over he oher s shown n Table 1. Table 1 Coss for he wo duraon alernaves a 1 a 2 a 3 a 4 a 5 a 6 Alernave Alernave Fgure 3 Nework dagram of he example (see onlne verson for colours) 4.2 Robus soluon vs. deermnsc soluon We evaluae and compare he performance of he robus and he deermnsc soluons when he unceran daa s seleced randomly. We solve he robus model for dfferen budges of uncerany. We hen generae 1000 random realsaons for all he cos and duraon coeffcens accordng o a normal dsrbuon on he one hand and a unform dsrbuon on he oher hand. In he case of he normal dsrbuon, we generae he values for he unceran coeffcen d from a dsrbuon wh mean d and sandard devaon 0.5d ˆ. In he case of he unform dsrbuon, he suppor s [ d ˆ, ˆ d d + d ]. Smlarly, c s generaed eher from a normal dsrbuon wh mean c and sandard devaon 0.5c ˆ, or from a unform dsrbuon on [ c, ˆ c + c]. We measure he performance of he robus and he deermnsc soluons n erms of he average realsed cos and he probably and amplude of he fnsh me consran volaons over he 1000 generaed realsaons. The probably of he fnsh me consran volaons s found as he number of mes he fnsh me consran s negave ou of he 1000 realsaons. The amplude s found as he average of he absolue value of he fnsh me consran volaon, every me s volaed. The npus we choose are:

13 Projec schedulng and resource allocaon 181 Γ1 = Γ 2 = Γ 3 = Γ 4 = Γ 5 = Γ 6 [0, 2], whle Γ 0 = 0 Γ 0 [0,14], whle Γ1 = Γ 2 = Γ 3 = Γ 4 = Γ 5 = Γ 6 = 0 where Γ 0 s he budge of uncerany of he cos coeffcens and Γ = Γ 1 = Γ 2 = Γ 3 = Γ 4 = Γ 5 = Γ 6 are he budges of uncerany for he duraon coeffcens of each ask. We carry ou he numercal sudy for d ˆ 0.1, ˆ = d c = 0.1c and d ˆ 0.5, ˆ = d c = 0.5 c. We observe no dfference n he performance of he robus soluon and he deermnsc soluon when uncerany s nroduced n only he cos coeffcens (Table 2). The cos assocaed wh alernave 2 s wce as ha of alernave 1, for every ask. For a range of uncerany of cˆ 0.1 or ˆ = c c = 0.5c, he maxmum cos for alernave 1 s sll much lower han he cos of he alernave 2. Hence, he resource allocaons of he asks and schedulng of asks does no change for cos uncerany n he robus model for any budge of uncerany Γ 0. Table 2 Average realsed coss for only cos uncerany Unform dsrbuon Normal dsrbuon Deermnsc ( cˆ = 0.1 c ) Robus ( cˆ = 0.1 c) Deermnsc ( cˆ = 0.5 c) Robus ( cˆ = 0.5 c ) When uncerany s nroduced n he duraon coeffcens, he robus ask schedulng s found o be dfferen from he deermnsc soluon for any budge of uncerany. The robus soluon s compued n a way ha for any value of he ask duraons whn he uncerany se, changng he soluon (reconfguraon) o avod consran volaons wll no be requred. The asks are scheduled n a way ha provdes he necessary buffer n each fnsh me consran so ha he sar me of a ask s greaer han or equal o he fnsh ask of he prevous ask even f he duraons vary whn he uncerany se (Fgures 4a 4c) and he projec can be compleed n 8 uns of me. In Fgures 4b and 4c, we observe ha n he robus soluon, ask a 1 s compleed n abou 1 un of me less han n he deermnsc soluon (Fgure 4a). Ths change n he duraon of ask a 1 calls for a shorer duraon alernave requrng more resources and hus ncreases he cos (Table 3). However, allows o creae buffer me necessary o avod he need of reconfguraon. When he range of uncerany n he coeffcens s ncreased, he buffer me needed wll also ncrease. In Table 4, for d ˆ = 0.5d, a budge as low as 0.4 requres shorer duraon alernaves for asks a 1 and a 6 n he robus allocaon when compared o he deermnsc allocaon, whereas, for d ˆ = 0.1d and any Γ only ask a 1 requres a shorer duraon alernave. Also, as he budge of uncerany ncreases, he soluon has o provde proecon agans bgger changes n a larger number of coeffcens and more asks requre shorer duraon alernaves (Table 4). Ths s because as he budge of uncerany ncreases, he allowed cumulave varaon of he unceran coeffcen ncreases. As he resource alernaves change, he cos also ncreases (Tables 5 and 6). In hs example, even hough he number of resources requred for he asks changed, he oal number of N and D resources requred n he sysem a any pon of me remaned a a maxmum of 2 each, so no addonal resources are hred.

14 182 E. Adda and P. Josh Fgure 4 Tme dagrams showng he execuon of he asks: (a) deermnsc, (b) robus, d ˆ = 0.1 d, Γ = 0.7 and (c) robus, d ˆ = 0.5 d, Γ = 0.7 (see onlne verson for colours) (a) (b) (c)

15 Projec schedulng and resource allocaon 183 In Fgures 5 8, we observe ha he robus soluon s beer proeced agans consran volaons han he deermnsc soluon as he robus approach provdes a buffer me o allow for he varaon n he coeffcens, whereas he deermnsc soluon s gh and a delay n a ask wll cause he projec o end lae. We observe ha when he range of uncerany s ncreased from d ˆ 0.1 o ˆ = d d = 0.5d, he amplude of volaon ncreases under boh probablsc dsrbuons as he varance of he realsed values s ncreased. However, he probably of consran volaon s smlar. Therefore when he range ncreases, consran volaons occur as ofen, bu when hey do occur hey are larger. We fnd ha he probably and amplude of consran volaons decreases wh an ncrease n he budge llusrang ha he robus model s beer proeced agans consran volaons as Γ ncreases. Ths s because as he budge of uncerany for he duraon coeffcens Γ ncreases, he soluon wll be robus o wder changes and n a larger number of coeffcens. Table 3 Resource requremens for d ˆ = 0.1d Tasks a 1 a 2 a 3 a 4 a 5 a 6 (N,D)-Deermnsc (1,0) (2,1) (2,1) (1,0) (2,1) (1,0) (N,D)-Robus (2,0) (2,1) (2,1) (1,0) (2,1) (1,0) Table 4 Resource requremens for d ˆ = 0.5d Tasks a 1 a 2 a 3 a 4 a 5 a 6 (N,D)-Deermnsc (1,0) (2,1) (2,1) (1,0) (2,1) (1,0) (N,D)-Robus (Budge [0,0.25]) (2,0) (2,1) (2,1) (1,0) (2,1) (1,0) (N,D)-Robus (Budge [0.35,0.65]) (2,0) (2,1) (2,1) (2,0) (2,1) (2,0) (N,D)-(Budge [0.7,2]) (2,0) (4,2) (4,2) (2,0) (2,1) (2,0) Table 5 Average realsed coss for dˆ = 0.1 d, cˆ = 0.1 c, Γ 0 = 0 Unform dsrbuon Normal dsrbuon Deermnsc soluon Robus soluon Table 6 Average realsed coss for dˆ = 0.5 d, cˆ = 0.5 c, Γ 0 = 0 Normal dsrbuon Unform dsrbuon Deermnsc soluon Robus (Γ [0, 0.25]) Robus (Γ [0.35, 0.65]) Robus (Γ [0.7, 2])

16 184 E. Adda and P. Josh Fgure 5 Consran volaons over a range of budges for ±10% range of uncerany ( dˆ = 0.1 d ), unform dsrbuon: (a) amplude of volaon and (b) probably of volaon (see onlne verson for colours) (a) (b)

17 Projec schedulng and resource allocaon 185 Fgure 6 Consran volaons over a range of budges for d ˆ = 0.1d, normal dsrbuon: (a) amplude of volaon and (b) probably of volaon (see onlne verson for colours) (a) (b)

18 186 E. Adda and P. Josh Fgure 7 Consran volaons over a range of budges for d ˆ = 0.5d, unform dsrbuon: (a) amplude of volaon and (b) probably of volaon (see onlne verson for colours) (a) (b)

19 Projec schedulng and resource allocaon 187 Fgure 8 Consran volaons over a range of budges for d ˆ = 0.5 d, normal dsrbuon: (a) amplude of volaon (b) probably of volaon (see onlne verson for colours) (a) (b) As seen above, uncerany n he duraons wll resul n ask fnsh me consran volaons (consran 5). Ths consran saes ha he fnsh me of a ask has o be greaer han or equal o s sar me plus s duraon. If he realsed duraon s longer han wha would be based on nomnal values, he nex ask wll be delayed. As a resul he projec may no be able o mee he gven deadlne (F) and volaes consran (6). We consder he followng wo scenaros for comparng he performance of he robus and he deermnsc soluons:

20 188 E. Adda and P. Josh he fnsh me deadlne can be volaed, ncurrng a penaly cos per me un he fnsh me deadlne mus be me and reconfguraon s necessary f he curren schedule does no allow o fnsh he projec on me. 4.3 Robus soluon vs. deermnsc soluon whou reconfguraon We compare he performance of he wo soluons n he scenaro where he oal fnsh me of he projec may go beyond he deadlne of 8 uns of me a an addonal cos. We nroduce a penaly cos per me un beyond he fnsh me deadlne. We fnd he new oal cos of he projec for he robus approach, and he deermnsc approach whou reconfguraon for dfferen penaly coss per un me and a range of budges. In Fgure 9, we see ha for no penaly cos (penaly cos = $0), he cos of usng he deermnsc approach s always lower han he cos of usng he robus approach as here s no penaly for volang he projec deadlne consran and he robus soluon corresponds o a resource allocaon wh hgher oal cos. However, when a penaly cos s ncurred for volang he fnsh me deadlne, he cos of usng he robus approach can be lower han he cos of usng he deermnsc approach. Ths s because he penaly cos ncurred due o consran volaon s lower for he robus approach han he deermnsc approach snce he deermnsc soluon may ncur projec deadlne consran volaon and hus a penaly. In he case where d ˆ = 0.1d, for penaly coss of $60 and above he oal cos of usng he robus soluon s always lower han he cos of usng he deermnsc soluon (see Fgures 9a and 9b). However, for penaly coss of $50 or less, he robus cos s hgher han he deermnsc cos a lower budges. Ths s because a low budges he realsed coeffcens vary moderaely from he nomnal values and hus he penaly amoun due o consran volaon s small and does no compensae he ncrease n he cos due o he change n he resource allocaon correspondng o he robus soluon. As he budge of uncerany vares, he resource allocaon changes once (for Γ = 0.1). For budges greaer han 0.1 he cos of he robus soluon s nondecreasng wh he budge of uncerany as a hgher budge mples less consran volaon and hus less penaly cos. In he case when d ˆ = 0.5d, as he budge of uncerany vares, he resource allocaon changes hree mes (Γ = 0.1, 0.35 and 0.7). For a gven resource allocaon, he robus cos decreases wh an ncrease n he budge of uncerany (see Fgures 9c and 9d), bu each change n he resource allocaon ncurs an ncrease n cos due o he selecon of a new duraon alernave wh hgher cos o creae more buffer me and allow more uncerany n he ask duraons, provdng beer proecon agans consran volaon. 4.4 Robus soluon vs. deermnsc soluon wh reconfguraon In applcaons where he fnsh me deadlne mus be me and volang he projec fnsh me by ncurrng a penaly cos s no an opon, reconfguraon becomes necessary. We llusrae ha he cos for he robus approach s lower han he cos ncurred n usng he deermnsc soluon and reconfgurng he projec when he acual progress s dfferen from he planned progress due o uncerany n duraon coeffcens.

21 Projec schedulng and resource allocaon 189 Fgure 9 Toal cos over a range of budges and varous penaly coss: (a) unform dsrbuon, dˆ = 0.1d, (b) normal dsrbuon, d ˆ = 0.1d, (c) unform dsrbuon, dˆ = 0.5d and (d) normal dsrbuon, d ˆ = 0.5d (see onlne verson for colours) (a) (b)

22 190 E. Adda and P. Josh Fgure 9 Toal cos over a range of budges and varous penaly coss: (a) unform dsrbuon, dˆ = 0.1d, (b) normal dsrbuon, d ˆ = 0.1d, (c) unform dsrbuon, dˆ = 0.5d and (d) normal dsrbuon, d ˆ = 0.5d (see onlne verson for colours) (connued) (c) (d)

23 Projec schedulng and resource allocaon 191 We consder all possble scenaros n whch he observed fnsh me of a varous asks s dfferen from he expeced fnsh me. When s observed ha he fnsh me deadlne (F) canno be aaned based on he curren schedulng and resource allocaon, we reconfgure o fnd a new schedule for he projec. As he deermnsc soluon s n general gh, n he sense ha here s no buffer me, he projec mus be reconfgured as soon as he ask endng lae forces he nex ask o sar lae. Reconfguraon mples selecng a shorer duraon alernave for a leas one of he asks lef a ha me. For example, f ask a 1 under he resource alernave (1,0) (cos = $5, nomnal duraon d 1j = 2) s delayed by dˆ 1 j = 0.5d 1 j = 1, hen he compued deermnsc soluon, whch gves a fnsh me of exacly 8 uns, s no longer vald as he projec fnsh me wll go beyond 8 uns of me due o he delay n ask a 1 of one me un. Thus, reconfguraon becomes necessary when ask a 1 has ended lae. Shorer duraon alernaves for a leas one of he remanng asks are requred o mee he deadlne. In hs example, reconfguraon leads o assgnng a shorer duraon alernave o asks a 4 and a 6 (assgnng wo resources nsead of one for boh asks, reducng he nomnal duraons o 1 un of me for each ask). Ths resuls n a cos ncrease of $10. If he robus soluon would have been used, here would be no need for reconfguraon, as he robus soluon s well prepared o handle hs change n he duraon of ask a 1. There would hus be no furher ncrease n cos (Table 7). Table 7 Resource requremens, oal cos for he deermnsc soluon wh reconfguraon and he robus soluon for d ˆ = 0.5d Tasks a 1 a 2 a 3 a 4 a 5 a 6 Toal cos (N,D)-Deermnsc (1,0) (2,1) (2,1) (1,0) (2,1) (1,0) $290 (N,D)-Reconfguraon (1,0) (2,1) (2,1) (2,0) (2,1) (2,0) $300 (N,D)-Robus (2,0) (2,1) (2,1) (1,0) (2,1) (1,0) $295 The change n duraon alernaves due o reconfguraon may no be he same as he robus soluon and he cos may be hgher. In he same way, for a delay n asks a 2 or a 3 he oal cos afer reconfguraon s also found o be $300 as he resource allocaons of asks a 4 and a 6 also have o be changed. A delay n ask a 4 whle mplemenng he deermnsc soluon wll no allow reconfguraon: he projec canno mee s fnsh me deadlne because ask a 4 s he las ask of he projec (see Fgure 4a), whereas he robus soluon can handle hs delay. Delays n asks a 5 or a 6 do no requre reconfguraon of he deermnsc soluon snce hese asks are scheduled o occur from me 5 o me 7, no ask s scheduled o sar afer her compleon, and he fnsh me deadlne s 8, allowng a delay of up o 1 un of me whou creang a consran volaon. Table 8 summarses he resuls. Table 8 Toal cos for he deermnsc soluon wh reconfguraon for dfferen scenaros Scenaro Toal cos afer reconfguraon Toal cos robus Delay n a 1 $300 $295 Delay n a 2 $300 $295 Delay n a 3 $300 $295 Delay n a 4 No reconfguraon possble, projec fals $295 Delay n a 5 $290 $295 Delay n a 6 $290 $295

24 192 E. Adda and P. Josh Thus for a delay less han 1 un of me n any of he asks, he average cos of reconfguraon s found o be $296 (dsregardng he scenaro when reconfguraon s no possble), whch s hgher han he cos of usng he robus soluon. 5 Conclusons In hs paper, we presened a robus opmsaon model for projec schedulng and resource allocaon n hghly unceran envronmens. The orgnal deermnsc model s a Per ne-based projec managemen sysem ha can be reconfgured o oban new schedules f he fnsh me deadlne of he projec canno be me due o uncerany n he projec parameers. We compared he performance of he deermnsc model and he formulaed robus model. We showed ha he robus model does no generally need reschedulng for a predefned level of uncerany. We llusraed ha he robus model provdes beer proecon agans consran volaons n he even of uncerany when compared o he deermnsc model a he expense of a small ncrease n cos. We showed ha he robus cos s lower han he deermnsc cos when volaon of he fnsh me deadlne of a projec ncurs a hgh enough penaly cos per me un of overme. We llusraed ha he oal cos of usng he robus approach may be lower han he cos of usng he deermnsc approach wh reconfguraon when exceedng he fnsh me deadlne s no an opon. References Adda, E. and Peraks, G. (2006) A robus opmzaon approach o dynamc prcng and nvenory conrol wh no backorders, Mahemacal Programmng, Vol. 107, Nos. 1 2, pp Adda, E. and Peraks, G. (2009) Dynamc prcng and nvenory conrol: robus vs. sochasc uncerany models, workng paper. Adda, E. and Peraks, G. (2009) Dynamc prcng and nvenory conrol: uncerany and compeon, Operaons Research (forhcomng). Angus, R.B., Gundersen, N.A. and Cullnane, T.P. (2000) Plannng, performng and conrollng projecs: prncples and applcaons, 2nd ed., Prence Hall, Upper Saddle Rver, NJ. Ayug, H., Lawley, M., McKay, K., Mohan, S. and Uzsoy, R. (2005) Execung producon schedules n he face of unceranes: a revew and some fuure drecons, European Journal of Operaonal Research, Vol. 161, No. 1, pp Ben-Tal, A., Golany, B., Nemrovsk, A. and Val, J.P. (2005) Realer-suppler flexble commmens conracs: a robus opmzaon approach, Manufacurng and Servce Operaons Managemen, Vol. 7, No. 3, pp Ben-Tal, A. and Nemrovsk, A. (1997) Robus russ opology desgn va semdefne programmng, SIAM Journal on Opmzaon, Vol. 7, No. 4, pp Ben-Tal, A. and Nemrovsk, A. (1998) Robus convex opmzaon, Mahemacs of Operaons Research, Vol. 23, pp Ben-Tal, A. and Nemrovsk, A. (1999) Robus soluons o unceran lnear programs, Operaons Research Leers, Vol. 25, pp Ben-Tal, A. and Nemrovsk, A. (2000) Robus soluons of lnear programmng problems conamnaed wh unceran daa, Mahemacal Programmng, Vol. 88, pp

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