Approximation Algorithms for Scheduling with Rejection on Two Unrelated Parallel Machines

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1 (ICS) Iteratoal oural of dvaced Comuter Scece ad lcatos Vol 6 No 05 romato lgorthms for Schedulg wth eecto o wo Urelated Parallel aches Feg Xahao Zhag Zega Ca College of Scece y Uversty y Shadog Cha bstract I ths aer we study the schedulg roblem wth reecto o two urelated arallel maches We may choose to reect some obs thus currg the corresodg ealty he goal s to mme the makesa lus the sum of the ealtes of the reected obs We frst formulate ths schedulg roblem to a teger rogram the rela t to a lear rogram From the otmal soluto to the lear rogram we obta the two algorthms usg the techue of lear rogrammg roudg I cocluso we reset a determstc -aromato algorthm ad a radomed - aromato algorthm for ths roblem Keywords Schedulg; eecto; romato algorthm; ear rogrammg; oudg I INODUCION he urelated arallel mache schedulg roblem to mme makesa C followg the otato of ma Graham et al [] s oe of the classc NP- hard roblems combatoral otmato hs roblem s metoed may works cocerg aromato algorthms [ ] ad t has receved much atteto the ast few decades wth may etat aromato algorthms amog whch the curretly best -aromato algorthm s due to [4] who also show that the roblem does ot admt a algorthm wth aromato rato smaller tha uless P = NP Oe secal case for ths roblem-ach ob ca oly be assged to a subset of the mache set wth the same rocessg tmewhch s also kow as the restrcted assgmet roblem Does there est a aromato algorthm wth aromato rato better tha? hs roblem s regarded as oe of the te oe roblems combatoral otmato [] he schedulg roblem wth reecto arses make-toorder roducto systems wth lmted roducto caacty ad tght delvery reuremets where smultaeous ob reecto ad schedulg decsos have to be made for mamg the total reveue I such systems accetg orders wthout cosderg ther mact o the whole system may delay some of the orders beyod ther due dates o be able to reserve the hgh ualty of servce to customers of acceted orders the maufacturer has to determe whch orders to accet ad how to schedule them to mame total reveue ay mortat results cocerg the arallel mache schedulg roblems wth reecto aear recet years I referece [5] gels et al develo some techues to desg aromato algorthm for the geeral roblem wth reecto her ma techue s to reduce a roblem wth reecto to a schedulg roblem wthout reecto based o the lear rogrammg roudg method Hoogevee Skutella ad Woegger [6] cosder the reemtve schedulg wth reecto ad the goal s to otme the reemtve makesa o the m arallel maches lus the sum of the ealtes of the reected obs hey rovde a comlete classfcato of these schedulg roblems o comlety ad aromablty O the varat wth a arbtrary umber of urelated maches whch s PX-hard they roose a 58-aromato algorthm for t oreover ther results for urelated maches may be carred over to the corresodg reemtve oe sho schedulg roblem wth reecto ad Yua [7] cosder several arallel mache schedulg roblems wth deteroratg obs ad reecto he obectve s to mme the schedulg cost of the acceted obs lus he total ealty of the reected obs hey roose two fully olyomal tme aromato schemes for the roblems uder cosderato I referece [8] Gerstl ad osheov study schedulg roblems wth reecto ad geeral osto-deedet rocessg tmes o detcal arallel maches ad they troduce effcet algorthms for the roblems whch ru m O tme (whch s olyomal for a gve umber of maches) here are may other mortat results ([9-]) I ths aer we study the followg roblem: We have two maches together wth a ob set he rocessg tme of ob o mache s We may choose ot to rocess ob thus currg a ealty he obectve s to otme the makesa for the rocessed obs lus the sum of the ealtes of the reected obs If we deote by the set of rocessed obs ad the set of reected obs the roblem may be eressed as re Cma We orgae ths aer as follows: I secto wo we gve a determstc -aromato algorthm for the roblem uder cosderato I secto hree a radomed - aromato algorthm s reseted for ths roblem I secto Four we gve some cocludg remarks wwwacsathesaorg 60 P a g e

2 (ICS) Iteratoal oural of dvaced Comuter Scece ad lcatos Vol 6 No 05 II -PPOXIION GOIH FO re C ma We troduce a decso varable for ob wth the followg meag: f ob 0 else s reected For each mache-ob ar ; we troduce a decso varable meag: f ob 0 otherwse wth the followg s rocessed o Based o the above otatos we formulate the schedulg roblem to a teger rogram m s t 0 ; 0 We frst rela teger rogram () to the followg lear rogram Here ma OP stads for the otmal obectve value for the schedulg roblem We aly the followg roudg rocedure to oudg Procedure Ste : Wheever s reected) 0 set Ste : Otherwse set 0 (ob s rocessed) Ste : Wheever rocessed o mache ) set (ob (ob s Ste 4: lse set (ob s rocessed o mache ) We ow aalye the ualty of the soluto obtaed by oudg Procedure We frst take a look at the makesa ma : : : here m s t 0 ; 0 We deote by the otmal soluto to lear rogram () Obvously we have OP Smlarly we have herefore we get s for the total ealty for the reected obs we have : : : So we have : f f OP wwwacsathesaorg 6 P a g e

3 (ICS) Iteratoal oural of dvaced Comuter Scece ad lcatos Vol 6 No 05 Here f ma s the aromato rato he value for f s mmed f ad oly f e the mmum value for f s Based o the dscussos above we roose a - aromato algorthm for re Cma lgorthm Ste : Formulate the schedulg roblem to a teger rogram () Ste : ela teger rogram () to lear rogram () Ste : Solve lear rogram () ad obta a otmal soluto Ste 4: Set : : : Ste 5: egulate : Ste 6: Process the obs mache a arbtrary order cotuously o heorem lgorthm s a -aromato algorthm for re Cma III NDOIZD -PPOXIION GOIH FO re C ma I secto wo we have a determstc rcle that determes the set ad I ths secto the rocessed obs are assged to mache wth some robablty We stll use rogram () as a lear rogram relaato for the schedulg roblem For the otmal soluto to lear rogram () we aly the followg roudg rocedure oudg Procedure ~ Ste : Wheever s reected) Ste : Otherwse set 0 0 (ob s assged to mache set (ob s rocessed) ob wth robablty I roudg rocedure the set of reected obs s a determstc set Whle the schedule formed by the obs the load o s a radomed oe We deote by the radomed schedule ad by the mache makesa for the schedule Obvously we have ma For rocessed ob ~ obvously we have thereby we get ~ : : Smlarly we have emma Proof: he otato stads for the radom evet that obs are assged to mache whle the obs are assged to mache Obvously we have P r ~ : : ~ : wwwacsathesaorg 6 P a g e

4 : Pr : Pr ma : : P r : Obvously we have comleted (ICS) Iteratoal oural of dvaced Comuter Scece ad lcatos Vol 6 No 05 he roof s From the dscussos above we have s metoed secto we have a uer boud for the total ealty of the reected obs : : : Hece we have : ma OP he aromato rato ma s mmed whe ad the mmum value s We gve a radomed -aromato algorthm for re C ma the followg way lgorthm Ste : Formulate the schedulg roblem to a teger rogram () Ste : ela teger rogram () to lear rogram () Ste : Solve lear rogram () ad obta a otmal soluto Ste 4: Set : : Ste 5: For ob satsfyg assg ob to wth robablty ~ mache Ste 6: Process the obs mache a arbtrary order cotuously o heorem lgorthm s a radomed -aromato algorthm for re Cma IV CONCUSION I ths aer we study the schedulg roblem wth reecto o two urelated arallel maches We may choose ot to rocess some obs thus currg the corresodg ealty he goal s to mme the makesa lus the sum of the ealtes of the reected obs We reset a determstc - aromato algorthm ad a radomed -aromato algorthm for ths roblem We obta the two algorthms usg the techue of lear rogrammg roudg CKNOWDGN he authors are thakful to the revewers for ther valuable suggestos hs work was suorted by the Chese Socety of ogstcs ad the Cha Federato of ogstcs ad Purchasg Proect (05CSK-99) the ogstcs eachg ad esearch eformato Proects for Chese Uverstes (ZW04048 ZW04049) the led athematcs hacemet Program of y Uversty ad the atoal college studets' ovato ad etrereeursh trag rogram ( ) FNCS [] Graham awler K estra ad H G ooy Ka Otmato ad aromato determstc seuecg ad schedulg: survey als of Dscrete athematcs vol [] V V Vara romato lgorthms Srger Berl 00 [] D P Wllamso D B Shmoys he Desg of romato lgorthms Cambrdge Uversty Press odo 0 [4] K estra D B Shmoys ad ardos romato algorthms for schedulg urelated arallel maches athematcal Programmg vol [5] D W gels D Karger S G Kollooulos S Seguta N Uma ad We echues for schedulg wth reecto oural of lgorthms vol [6] H Hoogevee Skutella ad G Woegger Preemtve schedulg wth reecto athematcal Programmg Ser B vol [7] S S ad Yua Parallel-mache schedulg wth deteroratg obs ad reecto heoretcal Comuter Scece vol [8] Gerstl ad G osheov Schedulg o arallel detcal maches wth ob-reecto ad osto-deedet rocessg tmes Iformato Processg etters vol wwwacsathesaorg 6 P a g e

5 (ICS) Iteratoal oural of dvaced Comuter Scece ad lcatos Vol 6 No 05 [9] D Shabtay N Gasar ad Kas survey o offle schedulg wth reecto oural of Schedulg vol [0] ste ad H Z Hader Ole schedulg wth reecto ad wthdrawal heoretcal Comuter Scece vol [] ste ad H Z Hader Preemtve ole schedulg wth reecto of ut obs o two uformly related maches oural of Schedulg vol [] F u C Ng ad Q Zhag Otmal algorthms for sglemache schedulg wth reecto to mme the makesa Iteratoal oural of Producto coomcs vol [] X Y Q Wag u ad ag Sem-ole schedulg o two detcal maches wth reecto oural of Combatoral Otmato vol wwwacsathesaorg 64 P a g e