Mixed Task Scheduling and Resource Allocation Problems

Transcription

1 Task schedulng and esouce allocaton 1 Mxed Task Schedulng and Resouce Allocaton Poblems Mae-José Huguet 1,2 and Pee Lopez 1 1 LAAS-CNRS, 7 av. du Colonel Roche F Toulouse cedex 4, Fance {huguet,lopez}@laas.f 2 Unvesté de Pau, Dept Infomatque F Pau, Fance Abstact. Ths pape addesses a mxed task schedulng and esouce allocaton poblem. Ths mxed poblem s successvely teated as a tempoal constaned poblem on the one hand and as a tme and esouce constaned poblem on the othe hand. In the second case we popose an ntegated appoach fo the popagaton of esouce allocaton constants and classcal task schedulng constants. These two vew-ponts lead to the desgn of two ndependent popagaton algothms. The elatve punng powe s evaluated on classcal schedulng nstances n whch esouce flexblty has been ntoduced. The expements show two man thngs: the tme and esouce based algothm obvously outpefoms the stctly tme-based algothm; out of all esouce allocaton mechansms, the most effcent s based on an enegetc easonng. 1 Intoducton The mxed task schedulng and esouce allocaton (n shot TSRA) poblem conssts of a set of jobs, a set 7 of tasks and a set 5 of enewable esouces. A pecedence elaton, denoted by p, goups togethe tasks n chans, each chan coespondng to a job. Tasks ae pefomed wthout nteupton by only one esouce of 5 and when two tasks ae pefomed on the same esouce they must be sequenced. A task! 7 s chaactezed by ts statng tme st, ts fnshng tme ft, and the set of allowed esouces 5 " 5. Fo a task, ts duaton depends on the esouce! 5 t s assgned to: hence t s denoted by p. We denoted by 7 the set of tasks to be pefomed on esouce and by 3 the set of tasks that may be pocessed on esouce. Ths pape moe patculaly concens Job Shop (o Flow Shop) poblems wth esouce allocaton constants, so-called job shop (flow shop) schedulng wth multpupose machnes [2, 7]. Some effcent algothms ae known fo each ndependent poblem (schedulng and allocaton) but they ae unable to each optmalty fo the mxed poblem: heustc appoaches ae nvestgated n [1, 3, 7]. Snce constantbased appoaches have been poved to be an effcent and flexble way fo tacklng schedulng poblems, an extenson takng account of allocaton constants seems a

2 2 Mae-José Huguet and Pee Lopez pomsng way of eseach. To ou knowledge, the only woks that nvestgated a constant-based appoach fo TSRA poblems have been poposed so fa n [6] and [10]. In [10], a tme wndow s assocated to each task fo each possble assgnment. Constant popagaton mechansms lead to the updatng of task tme-wndows whch may suppess some possble esouce assgnment. A specfc ule dedcated to esouce allocaton s poposed: esouces that may be allocated to seveal tasks ae aggegated n a cumulatve esouce. Fo nstance, let jk be thee tasks that may be assgned on esouce 1 o 2. In the dsjunctve case, no moe than two tasks can be smultaneously pefomed on the set of esouces ^ 1 2`. An aggegated esouce of capacty 2 s assocated to ths set and popagaton fo cumulatve schedulng s appled. In ths pape, as n [6], esouce allocaton s consdeed as pocessng tme constants. Some constant popagaton mechansms ae close to those developed n [10]; howeve specfc constant popagaton mechansms based on enegetc easonng [8] ae poposed. In elaton to [6], ths pape addesses new constant popagaton mechansms fo esouce allocaton and poposes expements on TSRA nstances. In the followng, we consde the TSRA poblem fom two ponts of vew: as a Tempoal Poblem, and as a Tme and Resouce Constaned Poblem. In each case, we pesent some constant popagaton mechansms to handle esouce allocaton constants and the ntegaton of these mechansms wth classcal constant popagaton fo task schedulng. The last pat concens some expements to compae the elatve punng powe of the poposed constant popagaton mechansms. 2 Task Schedulng and Resouce Allocaton as Tempoal Poblems 2.1 Constant Modelng In the poblem unde study we dstngush pue tme constants (lmt tmes and pecedences) and esouce constants (allocaton and shang). The esouce constants can easly be modeled as tempoal constants as well: an allocaton decson s elated to the pocessng tme of the task assgned on the esouce unde consdeaton; a shang constant yelds a necessay sequencng fo all pas assgned to a gven esouce. Let V = ^ st,ft!7` be the set of decson vaables. The set of constants s expessed by potental nequaltes x # j $ x bj ( b j! R): C j between two vaables x and $ Lmt tme constants: the statng tme (esp. fnshng tme) of a task s ncluded between an ealest tme est (esp. eft ) and a latest tme lst (esp. lft ): ( st $ x0 # est ) % ( x 0 $ st # $ lst ) whee x 0 denotes the tme ogn vaable. x j :

3 Task schedulng and esouce allocaton 3 $ Pecedence constants: n the outng of a job, p j (.e. task j may be pefomed afte task s completed) s modelled by: ( st ft # 0). j $ $ Resouce allocaton constants: fo nstance, fo a task such that ^ 1 ` 5 = 2 wth pocessng tmes n [ p p ] on 1 and n p p ] on 2, the duaton 1 1 [ 2 2 of task belongs to [ mn( p p ), ( p p )] : ( ft $ st # mn( p, p )) 1 2 % ( st $ ft # $ ( p, p )). Moeove f not allowed: ( st $ ft # $ p )& ( ft $ st # p ) p < p, values n ] p p [ ae 1 2 $ Resouce shang constants: each task needs one esouce to be pefomed and each esouce can pocess only one task a tme. Each pa of tasks ( j) assgned to a common esouce must be sequenced: ( st ft # 0) & ( st $ ft # 0) ' j! 7. j $ j 2.2 Constant Popagaton The pevous set of constants s dvded nto two sub-sets: lmt tmes and pecedences fom a conjunctve set of potental nequaltes (all constants must be satsfed) whle esouce constants fom a set of dsjunctve sets of potental nequaltes (at least one constant of each dsjunctve set must be satsfed). The conjunctve set of potental nequaltes may be epesented by an oented and labelled gaph G = ( X E) n whee the set of nodes X s assocated to the set of vaables V, and an edge fom the potental nequalty j x to x $ x # b exsts. j x j labelled by b j s n the set of edges E f Popagaton on the Conjunctve Set of Potental Inequaltes The Floyd-Washall (FW) algothm appled on a gaph G = ( X E) acheves 3- consstency by computng the longest path between each couple of nodes. It may then deduce the tghtest constants on the conjunctve set of potental nequaltes. Its tme 3 complexty s n O (_X_ ). Despte ts polynomal complexty, the FW algothm s too much tme consumng fo lage sze poblems; t s woth applyng the Bellman- Fod (BF) algothm when seachng fo lmt tmes updatng only. Indeed, BF algothm acheves 2-consstency n O (_ X E_ ). Howeve BF algothm do not handle bnay constants that may appea n the dsjunctve sets of potental nequaltes, patculaly task duatons and pecedences. Hence we povde an extenson, temed BF*, that ntegates ths patcula pocessng. It computes n 2 O (_X_ ) a longest paths between some pas of vaables va x 0.

4 4 Mae-José Huguet and Pee Lopez Popagaton on the Oveall Set of Constants To tackle the oveall set of constants, an algothm has been poposed n [5]. Ths Tempoal Constant Popagaton (TCP) algothm s stongly elated to the Uppe- Lowe Tghtenng algothm poposed n the TCSP aea [11]. It ntegates FW algothm and smplfcaton ules appled on the set of dsjunctve sets of potental nequaltes: a ule emoves a potental nequalty found nconsstent elatvely to the conjunctve constants; anothe ule emoves a dsjunctve set whch contans a potental nequalty aleady satsfed by conjunctve constants. Removng potental nequaltes may esolve some conflcts n esouce shang o suppess some nconsstent esouce allocatons. TCP and TCP* ae two algothms based on FW and BF* espectvely. They teate the pocedue on the conjunctve set and the smplfcaton ules on the set of dsjunctve sets untl an nconsstency s detected o no moe deducton can be deved. Howeve, snce esouce constants ae modelled by tempoal constants, the semantcs of the constants s fogotten and the algothm cannot consde the specfcty of TSRA (see secton 4). 3 Task Schedulng and Resouce Allocaton as Tme and Resouce Constaned Poblems In ths pat, we consde sepaately tme and esouce constants. Ou goal s to defne constant popagaton mechansms fo esouce allocaton and analyze the nteacton wth the popagatons obtaned on tme and shang constants. Two Dedcated Constant Popagaton algothms ae devsed. They ae based ethe on FW pocedue (DCP) o on BF* pocedue (DCP*). 3.1 Constant popagaton fo task schedulng We etan two classcal ules to detect fobdden pecedences between each pa of conflctng tasks. Rule FP-1 s based on tempoal easonng [4] and ule FP-2 s based on enegetc easonng [8]. $ Rule FP-1: fo j! 7, f lft j $ est < p + p then p/ j (.e. j p )., j, $ lft $ est + p + p coesponds to the length of the path fom The amount ( j ), j, st j to ft va ft j, x 0 and poduces an nconsstency wth the altenatve sequence sequencng s nfeasble. st n the conjunctve set of constants. If ths path p j then such a In the dsjunctve case, the enegy equed by on ove an nteval (, temed ( w, s gven by the ntesecton of ( wth the pocessng of. Ths enegy s mnmal when the pocessng of s ealzed fo postons that ovelap ( as less as possble.

5 Task schedulng and esouce allocaton 5 [ est lft j ] $ Rule FP-2: fo j! 7, f lft j $ est < p + p + ) w j l then p/ j j (.e. j p ). The eventual sequencng decsons obtaned wth FP-1 and FP-2 yeld adjustments of the elease date of and the due date of j. Rule FP-2 subsumes ule FP-1 (n dsjunctve case); convesely, FP-2 s much moe tme consumng than FP-1. Thus, n DCP algothms ule FP-1 s fst appled to suppess some pecedences between conflctng tasks and ule FP-2 s appled on the pecedences stll not poved nfeasble. 3.2 Constant popagaton fo esouce allocaton We popose fou ules, FA-1 up to FA-4, to deduce fobdden assgnments. $ Rules FA-1 and FA-2 ae espectvely based on FP-1 and FP-2 to detect nfeasble sequencng on a esouce fo a pa of tasks ( j) such that! 7 and j! 3 : f p/ j and j p/ then task j cannot be assgned to. As fo ules FP-1 and FP-2, FA-1 s fst appled to suppess some assgnments and ule FA-2 only consdes emanng undecded assgnments. $ Rule FA-3 suppesses assgnments havng an nconsstent duaton elatvely to the deduced duaton. Fo a task that may be pefomed on esouce wth duaton n [ p p ], let [ p p ] be the duaton deduced by tempoal constant popagaton, f [ p p ] + [ p,p ]=*, then task cannot be assgned to. $ Rule FA-4 consdes all the tasks n 7 and one task j of 3, f thee s an nteval ( such that W ( ( # ) w and ( ( ( l W < ) wl + w j then task j cannot be ( assgned to, whee W = ft( $ st( denotes the mal avalable enegy a dsjunctve esouce can povde on a gven tme nteval (. To smplfy the mplementaton of ule FA-4, we only consde the nteval ( coveng all the tasks: ( = [ mn ( estl ) ( lftl )] on whch the enegy consumpton of evey, j, j task l s ts pocessng tme. Rule FA-4 s then: f W ( # ) and W ( < ) p l + p j then task j cannot be assgned to. p l The mplementaton of ule FA-4 depends on the ode n whch tasks ae assgned to a esouce. Fo nstance, afte a task s added to a set 7, one may not fnd an nconsstency wheeas t would ase befoe addng that task. To pevent ths ptfall, we would have to consde, n the wost case, all subsets of 7 ncludng at least two

6 6 Mae-José Huguet and Pee Lopez tasks. In pactce, tasks n the subsets have to ntesect each othe and to ntesect wth the task consdeed n 3. In the DCP and DCP* algothms, ules ae teated untl a fx-pont s eached, that s ethe no moe deducton can be deved o an nconsstency s detected. The followng ode s selected n ou stategy: pocedue fo tme constant popagaton, ules dedcated to esouce allocaton (wth FA-1 befoe FA-2), and ules dedcated to esouce shang constants (wth FP-1 befoe FP-2). 4 Expements Ou expements concen Job Shop poblems wth esouce allocaton constants fom [7]. Set sdata coesponds to classcal Job Shop poblems (Fshe and Thompson, and Lawence nstances); n sets edata, data and vdata, altenatve esouces may pocess the tasks. In edata nstances, the aveage numbe of possble esouces s close to the ntal job-shop poblems; ths aveage numbe gows fo data and vdata nstances. We also consde Flow Shop ( FS ) poblems wth esouce allocaton constants fom [9, 12]. In all these benchmaks, the task duaton s fxed (fo each task on a esouce : p = p = p ) and s ndependent of the esouce allocated to pocess t (fo each task and fo each esouce! 5 : p = p ). Hence these nstances coespond to a smplfcaton of ou TSRA poblem defned n secton 1. In [7, 9, 12] the am s to detemne the mnmum value of the completon tme of jobs (makespan mnmzaton) denoted by C. Ou constant popagaton mechansms ae able to classfy a poblem as nconsstent (but cannot pove ts consstency); snce they do not fnd an optmal soluton, they ae embedded n a moe geneal pocedue that ams to compute the smallest value of job due-dates such that the popagaton mechansms do not detect an nconsstency. Ths value, denoted by C, s a lowe bound of C. To study the mpact of the poposed constant popagaton mechansms, we evaluate the ato between C and C : ( C $ C ) C. The smalle ths ato s, the moe poweful the constant popagaton ules. If C = C ou constant popagaton ules detect all nconsstences; f C < C some nconsstences stll eman n the nstances (the constant popagaton s not complete). Fo some nstances, the esults n [7, 9, 12] do not each the optmal value of LB LB LB makespan but a lowe bound, denoted by C. The ato ( C $ C ) C may be negatve when ou constant popagaton mpoves the lowe bound of the makespan. In the geneal pocedue, we test seveal combnatons of the poposed constant popagaton ules: $ TCP and TCP*: we apply the geneal tempoal constant popagaton ules (wth ethe FW o BF* fo the conjunctve pat of the tempoal constants).

7 Task schedulng and esouce allocaton 7 $ DCP and DCP*: n these combnatons, we apply ethe FW o BF* fo tme constant popagaton, all the ules fo esouce allocaton (FA-1 to FA-4), and ules FP-1 and FP-2 fo esouce shang. $ DCP wth only one ule fo esouce allocaton: that s, DCP/1 wth FA-1, DCP/2 wth FA-2, DCP/3 wth FA-3, and DCP/4 wth FA-4. One apples FW fo tme constant popagaton, and FP-1 and FP-2 fo the popagaton of esouce shang constants. In the followng tables, we summaze the esults gven by these combnatons. Fo sdata nstances, we dd not test the DCP/1 up to DCP/4 algothms snce they ae pue schedulng poblems. Column Tme (s) gves the aveage CPU tme n seconds to fnd a lowe bound of the makespan; column ato1 epesents the aveage of ) C LB C C ( C $ C (f aveage of LB C s known); column ato2 gves the postve ( $ ) C and the column ato3 povdes the negatve aveage of the same ato. In backets we ndcate the numbe of nstances fom whch we compute the coespondng ato. SDATA (43 nstances) EDATA (43 nstances) Algo. ato1 (%) Tme (s) ato1 (%) ato2 (%) ato3 (%) Tme (s) TCP (25) (17) (1) TCP* (25) (18) 0.00 (0) DCP (25) 0.76 (13) (5) DCP* (25) 1.44 (15) (3) RDATA (43 nstances) VDATA (43 nstances) Algo. ato1 (%) ato2 (%) ato3 (%) Tme (s) ato1 (%) ato2 (%) Tme (s) TCP (15) (25) (3) (26) (17) TCP* (15) (28) 0.00 (0) (26) (17) DCP (15) (24) (4) (26) (17) DCP* (15) (26) (2) (26) (17) FS [12] (169 nstances) FS [9] (63 nstances) Algo. ato1 (%) Tme (s) ato1 (%) Tme (s) TCP TCP* DCP DCP* These expements show a lage mpact of DCP elatvely to TCP n sdata, edata and Flow Shop benchmaks. In edata nstances fo ato1, C deduced by DCP s 2.3% away fom fom C C wheeas C deduced by TCP s 32.2% away. Moeove on these benchmaks DCP s usually faste than TCP. Howeve fo data and vdata nstances, DCP weakly mpoves the lowe bound of makespan deduced by TCP; n fact these nstances seem to be less had than edata nstances and constant popagaton does not deduce much nfomaton (few nconsstences ae detected). The compasons between TCP and TCP*, and between DCP and DCP* show a weak dffeence between the elatve powe of FW and BF*. Fo nstances n edata

8 8 Mae-José Huguet and Pee Lopez and ato1 : C obtaned by TCP s 32.21% away fom C s 42.57% away fom C ; and C wheeas fo TCP* C deduced by DCP s 2.3% away fom C wheeas 8.04% away fo DCP*. Howeve, the CPU tme consumed by BF* s much smalle than the consumpton of FW. It could possbly be nteestng to fst apply BF* and, when thee s no moe deducton, apply FW. EDATA (43 nstances) FS [12] (169 nstances) FS [9] (63 nstances) Algo. ato1 (%) ato2 (%) ato3 (%) Tme (s) ato1 (%) Tme (s) ato1 (%) Tme (s) DCP 2.30 (25) 0.76 (13) (5) DCP/ (25) 1.07 (16) (2) DCP/ (25) 1.07 (16) (2) DCP/ (25) 1.03 (17) (3) DCP/ (25) 0.81 (14) (4) Whateve the ule used n DCP/1 to DCP/4 fo the Flow Shop nstances, thee s no dffeence fo C deduced by DCP wth all ules. Actually n these nstances, the ules devoted to esouce allocaton do not un: all the deductons ae made by the popagaton of esouce shang constants. Moeove n 65.08% out of the Flow Shop nstances fom [9] and n 82.25% out of the examples fom [12], the value C obtaned by DCP s equal to the optmal makespan C. In edata nstances, the elatve powe of sole ule FA-1 (o FA-2) s poo elatvely to DCP wth all ules (n these nstances, thee s no dffeence between FA-1 and FA-2 wheeas FA-2 subsumes FA-1). The most effcent ule s FA-4 whch deduces by tself a lowe bound neae than the lowe bound deduced by DCP ncludng all the ules. To mpove ou expements on the elatve punng powe of ou ules, we have now to geneate ou own benchmaks wth unelated (and vaable) duatons fo unassgned tasks. Conclusons Ths pape pesents a constant-based appoach fo mxed task schedulng and esouce allocaton poblem. Ths poblem can be addessed ethe as a tempoal constaned poblem o as a tme and esouce constaned poblem. We popose some new constant popagaton mechansms to handle esouce allocaton and we ntegate these mechansms wth classcal constant popagaton pocesses fo tme constants and esouce shang constants. We pesent some constant popagaton mechansms that lead to the emovng of some task assgnments. The mplementaton of such mechansms stll needs to be mpoved to make the esults ndependent on the ode constants ae handled. Othe deducton pocesses have been defned to detemne nconsstent allocatons between pas of tasks and popagate these new constants. These constant popagaton ules have to be mplemented to evaluate the effcency. Moeove, as we dd not fnd n the lteatue benchmaks coespondng to ou TSRA hypotheses (vaable duatons and unelated duatons

9 Task schedulng and esouce allocaton 9 fo esouce allocaton) we have now to geneate new nstances to stctly evaluate the punng powe of ou ules. Refeences 1. Bandmate, P.: Routng and schedulng n a flexble Job Shop by Tabu seach. Annals of Ope. Res. 41 (1993) Bucke, P, Jush, B., Käme, A.: Complexty of schedulng poblems wth multpupose machnes. Annals of Ope. Res. 70 (1997) Dauzèe-Péès, S., Paull, J.: An ntegated appoach fo modelng and solvng the geneal multpocesso job shop schedulng poblem usng tabu seach. Annals of Ope. Res. 70 (1997) Eschle, J., Roubellat, F., Venhes, J-P.: Chaactezng the set of feasble sequences fo n jobs to be caed out on a sngle machne. Euop. J. of Ope. Res. 4 (1980) Esquol, P., Huguet, M-J., Lopez, P.: Modelng and managng dsjunctons n schedulng poblems. J. of Intellgent Manufactung 6 (1995) Huguet, M-J., Lopez, P.: An ntegated constant-based model fo task schedulng and esouce assgnment. CP-AI-OR-99, Feaa, Italy (1999) 7. Hunk, J., Jusch, B., Thole, M.: Tabu seach fo the Job Shop schedulng poblem wth multpupose machnes. OR Spektum 15 (1994) Lopez, P., Esquol, P.: Consstency enfocng n schedulng: a geneal fomulaton based on enegetc easonng. 5 th Int. Wokshop on Poject Management and Schedulng, Poznan, Poland, pp (1996) 9. Néon E.: Du Flow Shop Hybde au Poblème Cumulatf. Thèse de Doctoat, Unvesté de Technologe de Compègne (1999) 10.Nujten, W.: Tme and esouce constaned schedulng: A constant satsfacton appoach. PhD dssetaton, Endhoven Unvesty (1994) 11.Schwalb, E., Dechte, R.: Pocessng dsjunctons n tempoal constant netwoks. At. Intellgence 93 (1997) Vgne A.: Contbuton à la ésoluton des poblèmes d odonnancement de type monogamme, multmachne ( Flow Shop Hybde ). Thèse de Doctoat, Unvesté de Tous (1997)