Improved lower bounds for hard Project Scheduling instances

Transcription

1 Improved ower bounds for ard Proect Sceduing instances Guierme Henrique Ismae de Azevedo Universidade Federa Fuminense Rua Passo da Pátria, 56, saa 309 Boco D, São Domingos, Niterói - RJ guiermeen@gmai.com Artur Aves Pessoa Universidade Federa Fuminense Rua Passo da Pátria, 56, saa 309 Boco D, São Domingos, Niterói - RJ artur@producao.uff.br ABSTRACT In a Resource Constrained Proect Sceduing Probem wit generaized precedence restrictions (RCPSP/max), one must scedue a set of activities wit nown duration satisfying precedence restrictions wit variabe time ags and respecting te avaiabiity of resources. Te goa is to minimize te proect duration (maespan). Tis study presents a metod to cacuate a ower bound on te maespan based on constraint propagation tat considers tree reaxations of RCPSP/max. Te first and te second reaxations are soved by greedy metods and te ast one as a inear programming probem. We aso introduce artificia resources and oter preprocessing tecniques to improve te ower bound quaity. We report experiments using 58 open bencmar instances wit up to 200 activities were our procedure proved te optimaity of 3 nown soutions and cosed 62% of te optimaity gap on te average. It aso found better ower bounds for 26 out of 79 instances wit 500 activities. KEYWORDS. Proect Sceduing. Resource Constraint. Resource Constraint. Main area (inform by priority te área of te artice because JEMS system maes te cassification afabeticay) OC - Combinatoria Optimization PM - Matematica Programming AD & GP - OR in Administration & Production Management 24

2 . Introduction Optimizing proect panning is important in every nowedge area since it can avoid waste of time and resources, especiay in great proects ie rocet aunc, buiding ydroeectrica pants or oi and gas patforms (PINEDO, 2004). Te resource constrained proect sceduing probem (RCPSP) is a we-nown mode for suc optimization tat is defined as foows. Let V = { 0,,..., N, N + } be te set of activities to be scedued using a set R = {,, m} of resources. Eac activity V as an execution time p and uses a specific amount r i of resource i at every instant of its processing time. Resource i as R i units avaiabe during a te time orizon. Te scedue must aso respect activity precedence, wic are represented by te directed grap G = ( V, A). For eac (, ) A, we use d to denote te time ag from activity to. In tis case, as to start at east d time units after starts and as to start at most d units of time after starts. For te RCPSP/max, nonnegative cyces are aowed in A. Te activities 0 and N + represent te beginning and te end of te proect, aving nu execution times and nu use of resources. Every activity wit no oter predecessor wi succeed 0, and every activity wit no oter successor wi precede N +. In tis study, te obective is to find te minimum competion time for te proect, aso nown as maespan and aso defined as te finis time of activity N +. RCPSP and its variations are NP-ard as tey generaize sop probems, ie open sop or ob sop (BLAZEWICZ ET AL., 983). Since it is difficut and time costy to prove optimaity of soutions for tis probem, finding a good quaity ower bound (LB) may bring some usefu information during te decision maing process. Tere are two approaces in te iterature to cacuate a LB for te RCPSP: constructive and destructive metods. Constructive LBs use reaxed formuations of te probem, wic are, generay, easier to sove. One of te most used reaxations is te woroad based one. For a given resource i, te tota woroad required by eac activity is given by r i p. Te woroad based reaxation tries to scedue activities so as to satisfy te woroad requirements instead of te resource constraints. For exampe, Koné et a. (20) introduced a matematica formuation based on start and finis events. Kooi et a. (200) presented a Mixed Linear Probem (MIP) tat generaizes woroad constraints. Franc et a. (200) aso presented a woroad-based metod using te concepts of Baptiste et a. (999) and Dorndorf et a. (999). Scutt et a. (200) use a finite domain sover in a azy cause generation approac wit a ybrid of finite domain and Booean satisfiabiity soving. Demassey et a. (2005) define te destructive metod as a procedure tat defines ypotetica upper bounds T on te maespan aiming to prove tat tere is no soution respecting tis upper bound. If suc a proof is successfu, one can concude tat T + is a ower bound on te optima maespan. Brucer & Knust (2000) presented a destructive LB tat uses a constraint propagation metod and a MIP to try to prove tat T is an infeasibe upper bound. Te metod presented in tis paper contains tree different woroad approaces to be used in bot constructive and destructive LBs. Te first and te second ones are singe resource woroad reaxations of RCPSP/max. Te first reaxation does not consider te activities execution intervas, but te second does consider. Te tird metod uses a inear programming (LP) reaxation tat considers a resources in parae. At first, te tree metods are used in a constructive approac to update execution intervas and te LB on maespan. Once an interva is updated, te information is transmitted to te oter intervas troug by a precedence constraint propagation metod. After te constructive approac finises, we use te same tree reaxations in a destructive approac to update te LB on te maespan. We used te instances avaiabe at PSPLib (202) as bencmar. As Scutt et a. (200) proved optimaity for a great number of tem wit 0 to 200 activities in itte computationa 25

3 time, we consider ony te 58 ard instances tat coud not be soved by teir metod. A tese instances ave avaiabe feasibe soutions wose optimaity is not proved and avaiabe ower bounds reported by Franc et a. (200). For tese instances, no LB is provided by Scutt et a. (200). In te reported experiments, we proved te optimaity of 3 feasibe soutions for te first time, for instances wit 30 activities, and cosed 62% of te optimaity gaps on te average. We aso tested a faster but weaer version of our metod on 79 instances wit 500 activities, finding better LBs for 26 of tem, and cosing 5.5% of te optimaity gaps on te average. Tis paper as te foowing structure: Section 2 presents our approac to cacuate ower bounds on te maespan. Section 3 describes te impementation and Section 4 te tests and computationa resuts. Finay, Section 5 presents our concusions. 2. Our Approac Our metod maintains an execution interva for eac activity defined as an interva were it must be competey executed. Te execution interva of activity is denoted by ( ES, LF ), were te Eary-Start (ES) is te first instant it can start and te Late-Finis (LF) is te ast moment it can end. Atoug te execution intervas are not in te probem definition, tey are impied by in te precedence reations. To initiaize te execution intervas, first we consider ( ES, LF ) = ( 0, T ) for a V, were T is a vaid upper bound (UB). Ten we use te propagation metod, described beow, to cec and update tem if possibe. Tis metod is divided in two pases, one for ES and oter for LF updates. We propagate te ES updates as foows. Let us consider tat ES was updated. If tere is a precedence arc (, ) A tat does not respect te constraint ES + d ES, we update ES to ES + d. Tis cec must be repeated unti every updated ES is tested and no more updates are performed. Simiary, we propagate LF updates as foows. Wenever LF is updated, if te constraint LF LF d is not satisfied, due to (, ) A, we update LF to LF d. Tis cec must aso be repeated unti every updated LF is tested and no more updates are performed. Unti tis point, te execution intervas consider ony precedence reations. In te next subsections, we wi present te tree woroad metods to update te intervas considering aso te resource constraints, eac one based on a different reaxation of te probem. In te first reaxation, we scedue te woroad for eac resource individuay regardess te execution intervas. In te second, te woroad is aso scedue for eac resource individuay, but considering execution intervas. In order to increase te quaity of te ower bound, we aso present a way to incude artificia resources and oter tecniques to strengten te reaxation. Te tird metod is a domain reduction by LP tat considers a resources in parae, wic is stronger but sower tan te previous metods. For a te reaxations preemption is aowed. 2.. Domain reduction by singe-resource reaxation Te first metod presented to update te execution intervas uses a woroad based reaxation tat considers a singe resource to cec if activity V can start at ES and finis at LF. In order to cec te vaidity of te starting time ES for activity, we consider te tota woroad tat must occur before starts. If we prove tat it is not possibe to scedue a tis woroad in te interva ( 0, ES ), ten we can concude tat ES can be increased by at east one time unit. More generay, et α be an arbitrary instant before ES. If we prove tat it is not possibe to scedue te woroad tat must occur between α and te start of in te interva ( α, ES ), ten we can concude tat ES can be increased. Simiary, given an instant β after LF, if we find out tat it is not possibe to scedue a te woroad tat must occur between 26

4 te finis of and β in te interva (, β ) LF, ten we prove tat LF can be reduced by at east one time unit Woroad requirements Here, we sow ow to cacuate te woroad of eac V tat must occur in ( α, ES ) assuming tat starts at ES, or in ( LF, β ) assuming tat finises at LF. Beside te assumptions, te metod to cacuate te woroad for bot intervas is te same. Hence, we wi present it for a generic interva ( LB, UB) wit no assumption. Later, we sow ow to consider tese assumptions. Let ( L B, UB ) be te current execution interva for activity V and be te part of te processing time of tat must occur in te interva ( LB, UB). We can cacuate by te formua = p max{ φ ; Φ }, were φ is te part of te execution time of out of ( LB, UB) wen it is scedued starting at L B and Φ is te part of te execution time of out of ( LB, UB) wen it is scedued finising at U B. φ and Φ can be cacuated as foows: φ = max{ LB LB ;0} + max{ LB + p UB;0} () Φ = max{ UB UB;0} + max{ LB ( UB p );0} (2) Once we cacuated, te woroad of activity tat is required in te interva ( LB, UB) for te resource i is wi = r i. Wen cecing weter te activity V can start at ES, we do not use te current execution intervas for remaining activities. Instead, we temporariy update suc intervas assuming tat starts at ES and using te propagation metod described in te beginning of tis section. Ten, we cacuate te woroad of eac activity tat must occur in ( α, ES ) as sown above (et LB = α and UB = ES ) and use te singe resource woroad reaxation to cec if it is possibe to scedue a tis woroad for eac resource individuay. Simiary, to cec weter V can finis at LF we temporariy update te execution intervas assuming finises at LF and cacuate te woroad tat must occur in ( LF, β ) as sown above (et LB = LF and UB = β ). After a cecs are performed te execution intervas for a activities oter tan are restored to teir origina vaues regardess of weter te interva of is updated or not. If it is updated, we propagate te information to te oter execution intervas. For a given resource i, te tota woroad to scedue in ( α, ES ) is V { w \ } i and R α. Tus, in order to be possibe tat V to starts at te tota avaiabe wor is i ( ES ) ES, te constraint w ( α ) V { } i Ri ES \ updated to ( ) must be satisfied. Oterwise, ES can be α + w V { } i R \ i. To update LF, one coud use a simiar procedure, but LF, β and te constraint w R β. considering te woroad in te interva ( ) LF is updated to V i LF \ w V i R \ i ( ) ( ) { } i In tis case, if te constraint is not satisfied, β. { } We aso ca te update criterion described above singe-resource reaxation wit no execution interva, as it does not consider te execution interva of any activity for sceduing woroads. In te next subsection, we define te singe-resource reaxation considering te execution intervas and present a metod to sove it. Tis metod generaizes te ideas 27

5 introduced above to satisfy te interva woroad constraints. Te resuting reaxation is stronger but arder to sove tan te one wit no interva. We use bot reaxations in our metod Singe-resource woroad reaxation considering execution intervas Te singe-resource woroad probem (SRWP) can be described as foows. Let V = {, 2,..., N } be a set of activities to be scedued to a singe resource during te time orizon ( 0 ;T ) and aowing preemption. Eac V as a woroad w to be scedued during te time interva ( E S, LF ). Tere are H subintervas of te time orizon, eac subinterva starting at UB and finising at UB, for {,, H}. For every activity V, tere are subintervas and suc tat UB = ES and UB = LF. Te time orizon bounds are suc tat UB = 0 0 and UB H = T. Tere is a singe resource wit avaiabiity W on eac subinterva of te time orizon. Te obective is to find te amount of woroad for eac activity to be scedued in eac subinterva suc tat H V W for a {,, H} and = = w for a V. Assume witout oss of generaity tat L F LF LF 2 N. We use a greedy metod to sove te SRWP defined above. For every subinterva from to H, we scedue te activity wit owest index tat can be executed in ( ( E S, LF ) ( UB, UB ) ) and wose woroad is not competey scedued unti V W or tere is no activity tat satisfies te requirements to be scedued in. At te end, if every activity is competey H scedued, i.e. if te constraint u = u = w is satisfied for every V, ten te SRWP is feasibe. Oterwise, it is infeasibe. Next, we prove tat te previous procedure is correct. Lemma : Let S be a feasibe soution for te SRSP were tere is at east one pair of activities, V suc tat < and a part of is scedued in a subinterva before a part of wen bot activities coud be scedued. It is aways possibe to cange S so tat te woroad of is scedued in te same subinterva as or after, witout canging te scedue for te oter activities. Proof: In S, if V < W ten it is possibe to transfer part of to so tat and are in te same subinterva. If tere is sti a part of in ten te new and are in one of te foowing cases. If =, since bot and can be scedued in and LF LF, ten we can cange teir positions and S is sti feasibe as te tota woroad in and in do not cange. If > ten we divide in to ( ) = and ( ) 2 = ( ) to perform te cange between ( ) and. If < ten it is possibe to divide in to ( ) = and ( ) 2 = ( ) and perform te cange between ( ) and. In a cases, after te canges in S te tota woroad in and in do not cange, te sceduing for te oter activities woroad do not cange and te part of s woroad is in te same interva s or ater, as Lemma describes. Te expanation previousy presented sows tat for a given feasibe soution it is aways possibe to cange te sceduing order of 2 activities woroad, or parts of tem, so tat te activity wit te smaest LF is scedued before. Teorem : If te probem is feasibe, ten it is aways possibe to find a feasibe soution sceduing te activities in te eariest possibe time interva, respecting execution intervas and 28

6 woroad constraints. Proof: We prove it by induction on te vaue of N. For N =, te teorem obviousy ods. Assume tat te teorem ods for N =. We must prove tat it aso ods for N = +. Let = and S be a feasibe soution for tis probem suc tat tere is at east a part of an > scedued before a part of. By Lemma, it is possibe to perform a te necessary canges in order to ave competey scedued in te same interva or earier tan a te oter activities after ES. Tus, if te probem is feasibe ten te agoritm scedues in a feasibe position. For te remaining activities, consider modified probem witout were every woroad used by is subtracted from te avaiabiity of te corresponding intervas. By te inductive ypotesis, since tis subprobem as activities, te agoritm finds a feasibe soution for it if possibe. As a resut, it wi find a feasibe soution to te origina probem, wen it is feasibe Singe-resource woroad subprobem As te origina probem as m resources and te reaxation as a singe one, tere wi be m subprobems to cec te ES of eac activity, one for eac resource, and oter m subprobems to cec te LF of eac activity. To cec ES for resource i, tere wi be N = N activities, and eac activity V wi ave an associated V wit w = r i to be scedued in ( E S, LF ), were E S = max{ ES ;α} and L F = min{ LF ; ES }. Te wor avaiabe at te subinterva wi be W = Ri ( UB UB ). If at east one of te subprobems is infeasibe, ten ES can be updated to ES +. Tis procedure is used in a binary searc to define te new vaue of ES. After tat, te information is propagated to te oter activities of V. To cec LF for te resource i, we use a simiar procedure, but te activity intervas are defined as E S = max{ ES ; LF } and. L F = min{ LF ; β} If at east one of te subprobems is infeasibe, ten LF can be updated to LF. Atoug te singe-resource reaxation considers te execution intervas, some usefu information may be ost because te probem considers ony one resource at a time, witout precedence reations and te woroad constraint aows activities to use more resource units tan tey reay instanty require. In order to avoid it, we deveoped two additiona tecniques tat we present in te next subsection Artificia resources Te ower bounds obtained troug te woroad approac can be furter improved by te creation of artificia resources. Te effectiveness of suc tecnique reies on te fact tat te woroad reaxation aows for using resources fractionay in suc a way tat woud not be possibe in a feasibe soution. For instance, if two obs need two units of a given resource eac and tere are ony tree units avaiabe, te execution of tese two obs coud not overap in time. On te oter and, te woroad reaxation woud aow one ob and af of te oter to be executed simutaneousy. In tis case, instead of trying to consider te non-overapping constraint directy in te LB cacuation, we represent it as an additiona (artificia) resource wit one avaiabe unit and aving one unit required by eac ob, so tat it is taen into account witout canging te ower bounding agoritm. Te matematica concept of artificia resources is presented beow. Let r f be te requirement of activity for te artificia resource f and R f be te avaiabiity of tis resource during te time orizon of te proect. Te condition R f B, suc tat B is te optima vaue of te MIP (3)-(6), is sufficient to ensure tat te new resource does not cange te feasibe region of te RCPSP. 29

7 B = max r V f y (3) s.t.: r y R i R (4) V i + y i y (, ) A, d p or ES LF or ES LF (5) y 0, (6) { } In tis MIP, y is an auxiiary variabe tat is if is in te subset of activities tat maximizes te use of resource f, cacuated in equation (3), and 0 oterwise. Te cosen subset must respect te resource constraints and teir execution intervas must overap. Tose requirements are guaranteed by Equations (4) and (5), respectivey. In te next subsections, we present two specific famiies inds of artificia resources to be considered in te woroad metod. Te first is caed cique resources and te second, ypergrap cique resources Cique resources Let G = ( V, E) be an undirected grap were eac activity of V is a vertex and {, } E if and ony if and can not be executed in parae. A cique is a set of vertices Q V, suc tat if and, Q ten {, } E. For te construction of G, et, V be two activities. If for any resource i, r i + ri > Ri, if (, ) A wit d > p, if ES > LF or if ES > LF, ten {, } E. For eac cique Q, an artificia resource f can be added, wit r f = R f if Q and r f = 0 if Q. For simpicity, we consider R =. f Hypergrap cique resources Te main idea is to generaize cique resources to consider te cases were 3 activities can not be executed in parae, but pairs among tose 3 may be aowed. Let H = ( V, E ) be a ypergrap were eac activity of V is a vertex and eac yperedge E i = {,, } E if and ony if no more tan two activities among, and can be executed in parae. Since eac yperedge in E as exacty tree endpoints, H is cassified as 3-uniform. Eac subset Q V induces a sub-ypergrap H = ( Q, E ) of H suc tat E = { Ei E Ei Q }. A 3- uniform compete ypergrap H induced by Q in H is a sub ypergrap suc tat, for a,, Q, {,, } E, i.e. no more tan two activities in Q can be executed in parae. For te RCPSP, et te tree activities be,, V, if at east one of te edges {, }, {, } or {, } beongs to E as described in subsection 2.3., or for any resource i we ave r i + ri + ri > Ri. For eac generaized Q V tat induce a 3-uniform compete subypergrap in H, an artificia resource f is added wit r f = R f if Q and tere is no Q tat can be executed simutaneousy wit. If Q and tere is at east one Q tat can be executed simutaneousy wit, ten r f = R f 2. If Q ten r f = 0. For simpicity, we consider R = 2. f 2.4. Resource avaiabiity reduction Te LB obtained troug te woroad approac can aso be improved by reducing te resource avaiabiities witout canging te feasibe space wen it is possibe. For instance, consider a given interva of te proect time orizon, wose duration is 2 time units, were tree 30

8 units of a given resource is avaiabe, but ony two activities can be scedued eac one needing one resource unit, and aving a duration of 3 time units. It is cear tat, for te origina probem, te maximum resource usage at any instant of te given interva wi be 2. Notice tat, if tis information is not avaiabe to te woroad reaxation, it does not detect tat te two activities cannot be competey scedued in te given interva since te tota avaiabe wor is equa to te sum of teir woroads. On te oter and, if we consider tat te maximum resource usage is 2, tem te tota avaiabe wor wi be 4, forcing part of te activities to be scedued before or after tis interva. In a more genera way, et V t be te set of activities tat can be scedued in te interva t. Te maximum usage of te resource i can be cacuated by te MIP beow: max r i y (7) s.t.: V t r V i y Ri t y { 0, } Vt (8) (9) In tis MIP, y is a binary variabe tat is if te activity is in te combination tat maximizes te use of resource i and zero oterwise. One migt ave noted tat te previous probem is exacty te 0- napsac Probem. Tis probem is aso a NP-Hard probem, but it is widey studied in te iterature. In tis study, we considered te metod presented by Psinger (997) to sove it Domain reduction by inear programming Tis metod uses a inear programming reaxation to update eac activity bot ES and LF. Te idea is to consider te woroad for a te resources for every subinterva of te time orizon instead of considering te resource constraint. We use te reaxation defined beow. Let V = {, 2,..., N } 0 and U = {,..., T } be a set of activities to be scedued to m resources during te 0 be an ordered set of instants to divide te time orizon in subintervas. Eac activity requires r i units of te resource i and as an execution time p E S, L. Preemption is aowed. For every subinterva of te time time orizon ( ;T ) to be scedued in ( ) F orizon, te resource i as an avaiabiity of R i. Te tota woroad to i in must respect W i = Ri ( UB UB ) te avaiabe wor. Tere are H subintervas of te time orizon, eac subinterva starts at UB U and finises at UB U. For every activity V tere are UB, UB U suc tat UB = ES and UB = LF. Te obective is to define te maximum part of an activity tat can be scedued in ( a, b) for given instants a, b U. Te probem can be soved as te LP beow. ( ) Γ b α = max x =Γ( a) + (0) s.t.: H = V x = p r x R ( i UB UB ) { p ; UB UB } V () i R, =,, H (2) V i 0 x min =,, H, V (3) In tis LP, x represents te part of te activity tat is executed in te interva, V is te set of activities tat can execute in and Γ ( y) denotes te index of te interva tat 3

9 finises at y. Te equation (0) cacuates te maximum amount of s execution time tat can be scedued in ( a, b). Equation () guaranties tat a execution time of every activity is scedued. Equation (2) maes sure tat te woroad scedued for eac resource in eac subinterva does not exceed te avaiabe woroad. Equation (3) defines te domain of x. We use tis reaxation to define a new vaue to ES as foows. Let us consider V = V, m = m, ( 0, T ) = ( 0,T ), p = p, ( E S, LF ) = ( ES, LF ), r i = ri, R i = Ri a, b = ES, ES + p. For every activity, ES, ES + p, LF p, LF U. If and ( ) ( ) α < ten it means tat can not be competey scedued in ( ES ES + p ) p, for te reaxed probem, and so for te origina probem. Since te origina probem does not aow preemption, can not start before ES + ( p α ), tem ES can be updated to ES + ( p α ). To sove te reaxation and update LF, we use a simiar procedure. Te ony difference is tat ( a, b) = ( LF p, LF ). If α < p ten it means tat can not finis after LF ( p α ), tem LF can be updated to LF ( p α ). Since p = N + 0, it is not possibe to update te LB on te maespan using te procedure sown above to update ES N +. To update ES N + we use te same probem, but wit anoter obective. Te goa of te new probem wi be to find te minimum execution time scedued in ( a, b). Tis probem can be soved as te LP beow. α = minγ (4) s.t.: H = V x = p r x R ( i UB UB ) { p ; UB UB } V () i R, =,, H (2) V i 0 x min =,, H, V Γ b ( ) =Γ V ( a) + (3) γ x (5), = + to scedue a activities using te minimum amount of time after te current ower bound on te maespan. So, if γ > 0 ten we now it is not possibe to scedue a activities for te reaxed probem before ES + γ and so for te origina probem. As a resut, te LB on te maespan To update te LB on te maespan wit tis LP we define ( a b) ( ES, N T ) N + ES + γ. can be updated to N + We aso introduce te foowing cuts on demand to improve te LB quaity, were, UB UB. δ (, 2 ) denotes te execution time of tat must be scedued in ( ; ) 2 2 x = ( ; ) t = t δ ; 2 One migt ave noted tat reaxing te resource constraints to woroad ones may resut in te same oss of information mentioned in te subsection 2.3. So, in order to avoid some scedues tat are not aowed in te origina probem, we introduce woroad constraints for artificia resources on demand as te constraints of Equation (2). We use te same inds of (6) 32

10 artificia resources tat were presented in subsections 2.3. and Impementation Te concepts and reaxations presented before were used in two different metods: compete and simpified ones. Te compete metod starts initiaizes te ESs and LFs for every activity using te propagation metod presented in Section 2. After tat, it adds artificia resources described in 2.3. and unti every activity as r 0 for at east one artificia resource f. Next, it f appies te woroad reaxation wit no execution intervas and witout resource-avaiabiity reduction, as described in 2.. unti no more updates are possibe. Ten, te greedy metod described in 2..2 is executed wit resource-avaiabiity reduction described in 2.4. In te moment tat no more updates are possibe wit te greedy metod, we create and sove te LPs described in 2.5, introducing cuts on demand as previousy described. Once it is not possibe to update any interva using te current LPs, we use te propagation metod described in 2. If any ES or LF is updated, ten we go bac to te greedy metod iteration. Tis is repeated unti no more updates are found wit bot LP and greedy metods. Unti tis point, we ave used a constructive metod tat, not ony updates te LB on te maespan, but aso updates te execution intervas of te activities. Now, in order to furter improve te quaity of te LB on te maespan, we wi use te destructive process, described in Brucer and Knust (997). From now on, no update wi be vaid to ESs and LFs. Te process defines a T in ( ES, N + LFN + ) to be tested. Now, we use te two woroad-based reaxations previousy presented to update activity execution intervas. If at any moment we get ES + p > LF for any V ten tere is no feasibe soution wit maespan equa or ess tan T, i.e. T is an infeasibe vaue for te maespan. Te optima vaue for T in ( ES, N + LFN + ) is found troug a binary searc procedure. In order to compare te reaxations aone, we aso impemented te procedure considering ony te reaxations wit singe resource. We ca tis impementation simpified metod. Te woe procedure was impemented in C++ and a te tests were made in a 2.3 GHz Inte Core2Duo computer wit 4 GB of RAM. In order to sove LPs and MIPs, we use te CPLEX Computationa tests and resuts Te metods ere presented were tested in te 58 ard instances of te bencmar sets UBO50, UBO00, UBO 200, C, D and J30. Te ard instances are te ones tat Scutt et a. (200) coud not prove optimaity or found a LB. We aso sove te 79 bencmar instances of te set UBO500 tat are feasibe using te ony te simpified metod. Te LBs avaiabe at PSPLib (202) were used as bencmars to cec te quaity of our LBs. A instances are avaiabe at PSPLib (202). Tabe sows te resuts obtained by our compete metod for te 58 ard instances. In tis tabe te first coumn represent te set of instances tested, te second represent te number of instances tested in te set, coumns Same LB and Better LB represent te number of instances tat our metod coud find respectivey te same and a better LB tan te one avaiabe in te iterature, Cosed sows te number of instances wose optimaity was proved for te first time, Average Time (s) sows te average time (in seconds) to sove te instances, GAP% represents te average percentage GAP and % GAP cosed represent te percentage of te GAP cosed by our metod. 33

11 Tabe Resuts for compete metod Group Instances Same LB Better LB Cosed Average Time (s) GAP % % GAP cosed UBO % 66.99% UBO , % 24.40% UBO , % 0.05% C , % -4.78% D % -4.73% J % 8.6% Te LB found by tis metod is better in 62% of te instances and for oter 0% it is te same found in te iterature. Considering te set J30, te metod cosed 3 (50%) ard instances, found better LB for oter 2 (33%) and reduced te average percentage GAP in more tan 80%. For te cass UBO, we found better LBs for 87% of te ard instances and reduced te percentage GAP in more tat 0%. For te sets C and D, atoug te average GAP was not reduced, tis metod coud find better LB for 4 (24%) ard instances and te same avaiabe in te iterature for oter 5 (29%) of set C. Te Tabe 2 sows te resuts obtained by our simpified metod for te 58 ard instances and te 79 feasibe instances of te set UBO500. Tis tabe foows te same structure of Tabe. Tabe 2 Resuts for simpified metod Group Instances Same LB Better LB Cosed Average Time (s) GAP % % GAP cosed UBO % 66.99% UBO % 23.42% UBO , % 9.2% UBO , % 5.50% C % -0.3% D % -5.73% J % 8.6% Te LB found by tis metod is better in 43% of te instances and for oter 34% it is te same avaiabe in te iterature. Tis metod coud aso prove optimaity for 3 ard instances of set J30.Considering ony te ard instances of cass UBO, tis metod found better LB in 47% and reduced te GAP in more tem 5%. For UBO500 instances, tat Scutt et a. (200) did not tested, our metod coud find te same LB in 53% of te instances and found better LBs in 33%. For te sets C and D, te GAP is in average more tan 0% greater tan te one in te iterature. Comparing te compete and te simpified metods, it is cear tat te simpified one requires ess computationa effort, but te compete is important to reduce te GAP and to find better LBs for arger instances. 5. Concusion Considering te resuts obtained by te compete and te simpified metods presented in tis paper, atoug te LP reaxation is important to reduce te average percentage GAP, it is cear tat te compete metod requires muc more computationa effort. Comparing te resuts ere obtained and te ones avaiabe at PSPLib (202), we coud reduce te percentage GAP for a but te sets C and D. Te LBs were improved for 62% of te ard instances wit te compete metod and for 43% wit te simpified one. 34

12 Comparing te metod ere proposed wit te one presented by Scutt et a. (200), teir finite domain uses te Cumuative Sceduing Probem tat is simiar to te SRWP, but it does not aow preemption and te execution time is fixed. So teir reaxation is stronger tan te one used in tis wor. On te oter and, teir Booean satisfiabiity soving is used to add disunction restriction to te mode, eac disunction as a constraint. As te artificia resources introduced in tis wor are created considering more tan one disunction at a time, tey are stronger as tan te Booean satisfabiity. Anoter important difference between te metods is tat Scutt et a. (200) metod incudes a branc-and-bound procedure and te one proposed ere does not. A wea point of our metod is te fact tat it needs a vaid upper bound as input. Atoug te quaity of te upper bound does not cange te fina LB, it may affect te constructive LB and te tota execution time. Anoter important fact is tat our metod does not incude a euristic procedure, so if te nown upper bound is not optima ten we wi not find a better soution. On te oter and, te execution intervas obtained after te constructive part of our LB can be used in oter exact metods or in euristics. 6. References Baptiste, P; Pape, C. L.; Nuiten, W. and Pape, C. (999), Satisfiabiity Tests and TimeBound Adustments for Cumuative Sceduing Probems, Annas of Operations Researc, Bartusc, M.; Möring, R. H. and Radermacer, F. J. (988), Sceduing proect networs wit resource constraints and time windows, Annas of Operations Researc, Bazewicz, J.; Lenstra, J. K. and Kan, A. H. G. (983), Sceduing subect to resource constraints: cassification and compexity, Discrete Appied Matematics, 32, -24. Brucer, P.; Drex, A.; Moring, R.; Neumann, K. and Pesc, E. (999), Resourceconstrained proect sceduing: Notation, cassification, modes, and metods, European Journa of Operationa Researc, 2, 3-4. Brucer, P. and Knust, S. (2000), A inear programming and constraint propagation-based ower bound for te RCPSP, European Journa of Operationa Researc, 27, Demassey, S.; Artigues, C. and Miceon, P. (2005), Constraint-Propagation-Based Cutting Panes: An Appication to te Resource-Constrained Proect Sceduing Probem, Journa on Computing, 7, Dorndorf, U.; Pesc, E. and Pan-Huy, T. (2000), A time-oriented branc-and-bound agoritm for resource-constrained proect sceduing wit generaised precedence constraints, Management Science, 46, Franc, B.; Neumann, K. and Scwindt, C. (200), Truncated branc-and-bound, scedueconstruction, and scedue-improvement procedures for resource-constrained proect sceduing, OR Spectrum, 23, Koné, O.; Artigues, C.; Lopez, P. and Mongeau, M. (20), Event-based MILP modes for resource-constrained proect sceduing probems, Computers & Operations Researc, 38, 3-3. Kooi, A.; Haouari, M.; Hidri, L. and Néron, E. (200), IP-Based Energetic Reasoning for te Resource Constrained Proect Sceduing Probem, ISCO Internationa Symposium on Combinatoria Optimization, Pinedo, M, Panning and Sceduing in Manufacturing and Services, Springer, New Yor, Pisinger, D. (997), A Minima Agoritm for te 0- Knapsac Probem, Operations Researc, 45, Scutt, A.; Feydy, T.; Stucey, P. J. and Waace, M. G., Soving te Resource Constrained Proect Sceduing Probem wit Generaized Precedences by Lazy Cause Generation, Computer Science, Corne University, ttp://arxiv.org/abs/ v, 200. PSPLIB., Proect Sceduing Probem Library, ttp:// /pspib,