SEQUENCING JOBS WITH UNCERTAIN PROCESSING TIMES AND MINIMIZING THE WEIGHTED TOTAL FLOW TIME. Yuri Sotskov, Natalja Egorova

Transcription

1 88 Algothmc ad Mathematcal Foudatos of the Atfcal Itellgece SEQUENCING OBS WITH UNCERTAIN PROCESSING TIMES AND MINIMIZING THE WEIGHTED TOTAL FLOW TIME Yu Sotsov Natalja Egoova Astact: We cosde a uceta veso of the schedulg olem to seuece set of jos o a sgle mache th mmzg the eghted total flo tme ovded that ocessg tme of a jo ca tae o ay eal value fom the gve closed teval It s assumed that jo ocessg tme s uo adom vaale efoe the actual occuece of ths tme hee oalty dstuto of such a vaale etee the gve loe ad ue ouds s uo efoe schedulg We develo the domace elatos o a set of jos The ecessay ad suffcet codtos fo a jo domato may e tested olyomal tme of the ume of jos If thee s o a domato th some suset of set heustc ocedue to mmze the eghted total flo tme s used fo seuecg the jos fom such a suset The comutatoal exemets fo adomly geeated sgle-mache schedulg olems th 7 sho that the develoed domace elatos ae ute helful mmzg the eghted total flo tme of jos th uceta ocessg tmes Keyods: Schedulg oustess ad sestvty aalyss ACM Classfcato Keyods: F Noumecal algothms ad olems: Seuecg ad schedulg Cofeece: The ae s selected fom XIV th Iteatoal Cofeece "Koledge-Dalogue-Soluto" KDS 8 Vaa Bulgaa ue-uly 8 Itoducto Thee ae schedulg olems eal lfe hee jo ocessg tmes may e evaluated th hgh elalty efoe schedulg ad the vast majoty of academc eseach assumes that jo ocessg tmes ae ethe detemstc see oo [Taaev Sotsov Stusevch 994] ad the fst at of oo [Pedo 995] o adom vaales th o oalty dstutos the secod at of [Pedo 995] Hoeve t s ot ealstc to assume all the jo ocessg tmes have o oalty dstuto fo may othe actcal schedulg olems Fo the most schedulg evomets jo ocessg tmes ae uo vaales ad the oly fomato that ca e cetaly otaed efoe schedulg s aout loe ad ue ouds fo a jo ocessg tme As such a schedule otaed y assumg a ceta oalty dstuto may ot e close to the otmal schedule actcal ealzato of the ocess Due to ths easo methods of costucto of otmal ad aoxmate schedules ae actcally motat fo schedulg olems th uceta teval ocessg tmes [Kouvels Yu 997; Sotsov Sotsova 4] I ths ae e addess a schedulg olem he t s mossle to ota elale oalty dstutos fo the jo ocessg tmes Namely t s assumed that the ocessg tme of a jo ca tae ay eal value fom the gve teval of ucetaty egadless of the values tae y the ocessg tmes of othe jos Moe ecsely e cosde the o-eemtve sgle-mache seuecg olem th teval ocessg tmes to mmze the eghted sum of the jo comleto tmes The ae s ogazed as follos I the secod secto olem settg s gve The thd secto cotas a lteatue eve The foth secto emds a o-esult fo a sgle-mache schedulg olem th the fxed ocessg tmes ad the eghted total flo tme cteo The ffth secto cotas the ecessay ad suffcet codto ove hch a jo domates aothe oe e fo each set of ossle ocessg tmes thee exsts a otmal emutato th the same ode of these to jos A llustatve examle s gve the sxth secto Comutatoal esults fo adomly geeated staces th teval ocessg tmes ae gve the seveth secto The last secto esets a ef cocluso

2 Iteatoal Boo Sees "Ifomato Scece ad Comutg" 89 Polem Settg Thee ae jos { } to e ocessed o a sgle mache Fo each jo ostve eght > s gve Pocessg tme of a jo may tae ay eal value etee gve loe oud a ad ue oud a hch ae oly o efoe schedulg Real ume C s eual to the comleto tme of the jo ad cteo C C deotes mmzato of sum of the eghted comleto tmes of jos Let S {! } deote a set of all emutatos of jos fom the set { } By adotg the thee-feld otato α β γ toduced [Gaham et al 976] e deote the schedulg olem of seachg a otmal emutato th set S that mmzes the value C as a C A set T { : a } of vectos of the ocessg tmes s a ectagula ox the sace of o-egatve -dmesoal eal vectos If a vecto of the ocessg tmes s o efoe schedulg e a the olem a C ecomes covetoal olem C th the fxed jo ocessg tmes As t s ove [Smth 956] the latte olem ca e solved O log tme We call seuecg olem a C a uceta seuecg olem cotast to olem C called a detemstc oe Lteatue Reve ad Defto I case of the uceta olem a C thee may ot exsts a uue schedule that emas otmal fo all ossle ealzatos of the jo ocessg tmes Theefoe [Daels Kouvels 995] so-called oust schedule mmzg the ost-case asolute o elatve devato fom otmalty called ost-case eget as oosed to hedge agast ocessg tme ucetaty I [Daels Kouvels 995; Yag Yu ] uceta olem a C th mmzg total flo tme e t as assumed that fo each jo has ee cosdeed I [Aveah ; Aveah ; Daels Kouvels 995; Yag Yu ; Leedev Aveah 6] alog th cotuous tevals of ossle ocessg tmes defed y the aove set T the ocessg tme ucetaty as desced though a fte dscete set T T h of ossle ocessg tme vectos called sceaos Each sceao T eesets fxed ocessg tmes fo jo set hch ca e ealzed th some ostve ut uo efoe schedulg oalty Fo a secfc sceao T detemstc olem C ases hch ca e solved usg otmal jo emutato defed due to the follog SPT ule: Sot the jos accodg to o-deceasg ode of the ocessg tmes Whle detemstc olem C s comutatoally smle fdg a emutato hch mmzes the ostcase eget to the uceta couteat th dscete set of ossle sceaos T T h s comutatoally had olem Eg [Daels Kouvels 995] t as ove that to fd a emutato S mmzg the ost-case asolute eget s ay NP-had olem see [Gaey ohso 979] fo defto eve fo to ossle sceaos: h I [Yag Yu ] t as ove that to fd a emutato S mmzg the ost-case elatve eget s ay NP-had olem fo to ossle sceaos as ell I [Yag Yu ] t as ove that to fd a emutato S mmzg the ost-case asolute elatve eget s uay NP-had olem [Gaey ohso 979] fo uouded ume h of ossle sceaos Wost-case eget s also defed fo the ocessg tme ucetaty desced though a ectagula ox T of ossle vectos I [Leedev Aveah 6] t as ove that mmzg the ost-case asolute eget fo

3 9 Algothmc ad Mathematcal Foudatos of the Atfcal Itellgece olem a C s ay NP-had olem f tevals of the ocessg tmes have the same cete fo all jos I [Aveah ] t as sho y a examle that thee s o dect elatosh etee the comlexty of the uceta olem th the gve fte dscete set of ossle sceaos T T h ad the comlexty of the uceta olem th the gve set T of cotuous tevals of ossle sceaos Summazg ths oveve e ca oseve that fo the most classcal olyomally solvale detemstc schedulg olems the uceta couteats th the ost-case eget cteo ecome ay o uay NPhad olems I fact eve fo exstece of oly to sceaos of ossle ocessg tmes h to mmze the asolute o elatve eget mles a tme-cosumg seach ove set S of! emutatos of jos I ode to ovecome ths comutatoal comlexty some secal cases e oose to use seachg the mmal set of domat schedules emutatos toduced [La Sotsov 999] fo solvg the uceta jo-sho olem a C th the maesa ojectve fucto: C max{ C : } max Defto : Set of emutatos schedules ST S s a mmal domat set fo the uceta olem α a γ f fo ay vecto T set ST cotas at least oe emutato schedule hch s otmal fo the detemstc olem α γ th vecto of the jo ocessg tmes ovded that ay oe suset of set ST loses such a oety A mmal domat set ST as vestgated [Allahved Sotsov 3; Allahved Aldoasa Sotsov 3; La Sotsov 999; Leshcheo Sotsov 7] fo the maesa cteo C max ad [Allahved Aldoasa Sotsov 3; Sotsov Allahved La 4] fo the total flo tme cteo C I atcula o of [Sotsov Allahved La 4] as addessed to the total flo tme a to-mache flo-sho th the teval ocessg tmes: F a Cmax A geometcal algothm has ee develoed fo solvg the flo-sho olem Fm a C th m maches ad to jos Fo uceta flo-sho olems th to o thee maches suffcet codtos have ee detfed he the tasosto of to jos mmzes the total flo tme Wo of [Allahved Aldoasa Sotsov 3] as addessed to the case of seaate setu tmes th the cteo C max o C Namely the ocessg tmes ee fxed hle each setu tme as elaxed to e a dstuto-fee adom vaale th gve loe ad ue ouds Domace elatos have ee detfed fo a uceta flo-sho olem th to maches I [Allahved Sotsov 3] fo a to-mache flo-sho olem F a Cmax suffcet codtos have ee detfed he the tasosto of to jos mmzes the maesa C max I [Leshcheo Sotsov 7] the ecessay ad suffcet codtos ee used fo the case he a sgle schedule domates all the othes ad the ecessay ad suffcet codtos ee used fo the case he t s ossle to fx the otmal ode of to jos fo the maesa cteo C th teval jo ocessg tmes max The fomula fo calculatg stalty adus of a otmal schedule e the lagest value of deedet vaatos of the jo ocessg tmes fo the schedule to ema otmal has ee ovded [Sotsov Sotsova Wee 997] fo a jo-sho olem m a Cmax th m maches Stalty adus of a otmal schedule as vestgated fo olem m a Cmax [La Sotsov 999; Sotsov Wagelmas Wee 998] ad fo olem m a C [Basel Sotsov Wee 996] I cotast to efeeces [Basel Sotsov Wee 996; La Sotsov 999; Sotsov Sotsova Wee 997; Sotsov Wagelmas Wee 998] hee exoetal algothms ased o exhaustg eumeato of the sem-actve schedules see 84 [Taaev Sotsov Stusevch 998] ee deved fo costuctg mmal domat set ST fo uceta jo-sho olems ths ae e sho ho to fd set ST fo the olem a C olyomal tme Next e eset a auxlay esult fo the detemstc olem C max

4 Iteatoal Boo Sees "Ifomato Scece ad Comutg" 9 Detemstc Seuecg Polem I [Smth 956] t as ove that olem C ca e solved log O tme due to the follog suffcet codto fo otmalty of emutato S : It s easy to ove that eualtes ae also ecessay codtos fo otmalty of emutato S fo the olem C Theoem : Pemutato S s otmal fo the olem C f ad oly f eualtes hold Poof: Suffcecy of codto fo otmalty of emutato as ove [Smth 956] Next e ove ecessty of codto fo otmalty of emutato y cotadcto method Let emutato S e otmal fo the olem C Hoeve fo the latte emutato at least oe eualty fom codto s volated eg e assume that the follog ooste eualty holds: < hee } { Let us cosde emutato S hch defes fom emutato y tasosto of jos ad We ota the follog eualtes ovded that otato C s used: Let us calculate the dffeece of the ojectve fucto values defed fo emutato ad emutato : Thus the follog eualty holds: If e multly oth left-had sde ad ght-had sde of the latte eualty y facto e ota eualtes < ad > hch mles: < The latte eualty cotadcts to the aove assumto

5 9 Algothmc ad Mathematcal Foudatos of the Atfcal Itellgece that emutato s otmal fo the olem C The cotadcto otaed mles the ecessty of codto fo otmalty of emutato fo the olem C Theoem s ove Uceta Seuecg Polem Seach of the mmal domat set ST fo a uceta olem a C may e ased o costuctg a domace elato o the set of jos To ths ed e defe a domace elato as follos Defto : o u domates jo v th esect to T e u v f thee exsts a mmal domat set ST fo the olem a C that each emutato fom set ST has ethe the fom u v o the fom u v e a emutato ST jo u ecedes jo v Fom Defto t follos that mmal domat set ST fo the detemstc olem C s a sgleto: { } S T As a esult the follog domace elatos hold: Fo a geeal case of the olem a C the follog clam may e ove usg Theoem Theoem : Fo the olem a C jo u domates jo v th esect to T f ad oly f the follog eualty holds: u v 3 u av Due to Theoem f jo u domates jo v ad jo v domates jo the jo u domates jo as ell Thus domace elato u v s tastve Theoem allos us to fd a mmal domat set ST fo the uceta olem a C ad to eset set ST comact fom Ideed va checg codto 3 fo each a of jos u ad v fom the set e costuct dgah G A of domace elato o the set : Ac u v elogs to set A f ad oly f domace elato u v holds Ovously t taes O tme to costuct dgah G A If due to Theoem lealy odeed set of jos ll e costucted the set ST fo the olem a C ll e a sgleto: { } S T Ad emutato S ll e otmal fo ay ossle sceao T It s easy to covce that the case of { } S T eualty A must hold fo the dgah G A Illustatve Examle Let ut data fo the stace of the olem a C e gve colums 4 of Tale Tale Iut data fo the olem a C a / a /

6 Iteatoal Boo Sees "Ifomato Scece ad Comutg" 93 Va testg codto 3 of Theoem fo each a of jos 6 6 ; a ; a3 5 6 ; 5 a ; a4 u ad v e ota the follog elatos: ; 5 a7 3 5 ; 3 a a ; 4 a5 Thus codto 3 holds fo the follog odeed a of jos: ad ; ad 3 ; ad 4 ; 3 ad 5 ; 4 ad 5 ; 5 ad 6 ; 5 ad 7 ; 5 ad 8 Due to Theoem the follog domace elatos hold: jo domates jo ; jo domates jos 3 ad 4 ; jo 3 domates jo 5 ; jo 4 domates jo 5 ; jo 5 domates jos 6 7 ad 8 It s easy to vefy that thee ae o othe domace elatos excet those that ae tastve to the aove oes Theefoe mmal domat set S T fo ths stace of the olem a C cossts of! 3! emutatos Fg Dgah G A thout tastve acs Dgah G A defg set S T { } s eeseted Fg fo smlcty the tastve acs ae omtted Thus hle seachg otmal emutato fo ths stace of the uceta olem a C t s suffcet to test oly emutatos stead of 8! 56 feasle oes Comutatoal Results I ths secto e desce the testg of adomly geeated olems a C ad ase y exemets o PC the uesto of ho may as of jos fom set satsfy codto 3 ad ho lage eos of the otmal values of the cteo ae fo the schedules costucted usg dgah G A C The comutatoal algothm as coded C If elato u v as fulflled ovded that u < v ou algothm dd ot tested the valdty of the ooste elato: v u Theefoe a otmal emutato as otaed thout fal f eualty A as fulflled fo the costucted dgah G A Fo the exemets e used a AMD 3 MHz ocesso th 4 MB ma memoy We tested adom staces of the uceta olem a C th the follog umes of jos: { } The gve tege loe ad ue ouds of the ossle tege ocessg tmes ee ufomly dstuted the age [ ] We tested the follog eos L% of the uceta jo ocessg tmes: L { } Fo each jo the gve loe oud of a jo ocessg tme as adomly geeated the age [ ] ad the ue oud as comuted as follos: a L% /% Fo each jo the eght > as adomly geeated the age [ 5] Tale eesets the comutatoal esults fo 8 sees of the adomly geeated staces Each sees cluded staces th the same comato of the aove ad L The left-had sde of Tale colums

7 94 Algothmc ad Mathematcal Foudatos of the Atfcal Itellgece 6 eesets the comutatoal esults fo staces th the umes of jos fom set { } The ght-had sde of Tale colums 7 eesets the comutatoal esults fo staces th the ume of jos fom set {35 4 7} The ume of sees s gve colum fo sees umeed fom to 4 ad colum 7 fo sees umeed fom 4 to 8 The ume of jos oe stace s gve the coesodg colum o 8 The eo L of the uceta jo ocessg tmes ecetage s gve the coesodg colums 3 o 9 The aveage eo of the ojectve fucto value C calculated fo the heustc schedules costucted due to Theoem ad dgah G A th esect to otmal ojectve fucto value C s gve the coesodg colums 4 o amely values : ae gve colums 4 ad The aveage elatve ume of acs A ecetage costucted due to valdty of codto 3 th esect to the ume of acs the comlete ccut-fee dgah s gve the coesodg colums 5 o amely values A : % ae gve colums 5 ad The aveage CPU-tme secods used y the ocesso AMD 3 MHz fo solvg oe stace aoxmately o exactly s gve the coesodg colums 6 o Fom the exemets t follos that domace elato stated Theoem allo us to solve exactly all the staces fom the sees th umes ad 4 see colum 5 The loest elatve ume of acs e 78869% as costucted fo the sees th ume 75 The loest aveage ualty of the schedules e : as otaed fo the sees th the lagest ume 8 The lagest CPU-tme 37 s as otaed fo the sees th ume 48 The aveage ualty of the schedules otaed deeds of the eo L of the jo ocessg tmes ad emas almost the same fo the staces th dffeet ume of jos ovded that they have the same eo L% of the uceta ocessg tmes Iceasg smultaeously oth umes ad L deceases the ume of staces solved exactly due to Theoem Tale Comutatoal esults fo adomly geeated staces a C Eo L % Ojectve eo Nume of acs% CPUtmes Eo L % Ojectve eo Nume of acs% CPUtmes

8 Iteatoal Boo Sees "Ifomato Scece ad Comutg" Cocluso The ma ssue of ths ae s to sho ho to costuct a mmal domat set ST olyomal tme va costuctg dgah G A as a comact esetato of set ST of domat emutatos We estmated a stegth of usg mmal domat set ST y extesve comutatoal exemets fo adomly geeated olems a C th ume of jos fom the age [ 7] Blogahy [Allahved Sotsov 3] A Allahved YuN Sotsov To-mache flosho mmum-legth schedulg th adom ad ouded ocessg tmes Iteatoal Tasactos Oeatoal Reseach [Allahved Aldoasa Sotsov 3] A Allahved T Aldoasa YuN Sotsov To-mache flosho schedulg olem to mmze maesa o total comleto tme th adom ad ouded setu tmes Iteatoal Tasactos of Mathematcal Sceces [Aveah ] I Aveah Mmax eget solutos fo mmax otmzato olems th ucetaty Oeatos Reseach Lettes [Aveah ] I Aveah O the comlexty of a class of comatoal otmzato olems th ucetaty Mathematcal Pogammg Sees A [Basel Sotsov Wee 996] H-M Basel Yu N Sotsov F Wee Stalty of a schedule mmzg mea flo tme Mathematcal ad Comute Modellg [Daels Kouvels 995] RL Daels P Kouvels Roust schedulg to hedge agast ocessg tme ucetaty sgle-stage oducto Maagemet Scece

9 96 Algothmc ad Mathematcal Foudatos of the Atfcal Itellgece [Gaham et al 976] RL Gaham EL Lale K Lesta AHG Rooy Ka Otmzato ad aoxmato detemstc seuecg ad schedulg: A suvey Aals of Dscete Mathematcs [Gaey ohso 979] MR Gaey DS ohso Comutes ad Itactalty A Gude to the Theoy of NP- Comleteess Sa Facso: Feema USA 979 [Kouvels Yu 997] P Kouvels G Yu Roust Dscete Otmzato ad Its Alcatos Bosto: Klue Academc Pulshes USA 997 [Leedev Aveah 6] V Leedev I Aveah Comlexty of mmzg the total flo tme th teval data ad mmax eget cteo Dscete Aled Mathematcs [Leshcheo Sotsov 7] N Leshcheo Yu N Sotsov Realzato of a otmal schedule fo the to-mache flosho th teval jo ocessg tmes Iteatoal oual Ifomato Theoy & Alcatos [La Sotsov 999] T-C La Yu N Sotsov Seuecg th uceta umecal data fo maesa mmzato oual of the Oeatoal Reseach Socety [Pedo 995] M Pedo Schedulg: Theoy Algothms ad Systems Petce-Hall: Eleood Clffs USA 995 [Smth 956] WE Smth Vaous otmzes fo sgle-stage oducto Naval Reseach ad Logstcs Quately [Sotsov Allahved La 4] YuN Sotsov A Allahved T-C La Flosho schedulg olem to mmze total comleto tme th adom ad ouded ocessg tmes oual of the Oeatoal Reseach Socety [Sotsov Sotsova 4] Yu N Sotsov N Sotsova Schedulg Theoy: Systems th Uceta Numecal Paametes Ms: Uted Isttute of Ifomatcs Polems Belaus 4 Russa [Sotsov Sotsova Wee 997] Yu N Sotsov N Sotsova F Wee Stalty of a otmal schedule a jo sho Omega - oual of Oeatoal Reseach [Sotsov Wagelmas Wee 998] YuN Sotsov APM Wagelmas F Wee O the calculato of the stalty adus of a otmal o a aoxmate schedule Aals of Oeatos Reseach [Taaev Sotsov Stusevch 994] VS Taaev YuN Sotsov VA Stusevch Schedulg Theoy: Mult-Stage Systems Dodecht: Klue Academc Pulshes Nethelads 994 [Yag Yu ] Yag G Yu O the oust sgle mache schedulg olem oual of Comatoal Otmzato Authos' Ifomato Yu N Sotsov Pofesso DSc PhD Uted Isttute of Ifomatcs Polems of the Natoal Academy of Sceces of Belaus Sugaova st 6 Ms Belaus; e-mal: sotsov@emaas-ety Natalja G Egoova uo eseache Uted Isttute of Ifomatcs Polems of the Natoal Academy of Sceces of Belaus Sugaova st 6 Ms Belaus; e-mal: egoova@emaas-ety