Minimizing the Total Late Work on an Unbounded Batch Machine

Transcription

1 The 7h Ieraioal Symposium o Operaios Research ad Is Applicaios (ISORA 08) Lijiag, Chia, Ocober 31 Novemver 3, 2008 Copyrigh 2008 ORSC & APORC, pp Miimizig he Toal Lae Work o a Ubouded Bach Machie Jiafeg Re 1, Yuzhog Zhag 1 Suguo 2 1 College of Operaios Research ad Maageme Scieces, Qufu Normal Uiversiy, Shagdog, , Chia 2 Deparme of Mahemaics, Qufu Normal Uiversiy, Shagdog, , Chia Absrac We cosider he problem of miimizig he oal lae work ( V j ) o a ubouded bach processig machie, where V j = mi{t j, p j } ad T j = max{c j d j,0}. The bach processig machie ca process up o B (B ) jobs simulaeously. The jobs ha are processed ogeher form a bach, ad all jobs i a bach sar ad complee a he same ime, respecively. For a bach of jobs, he processig ime of he bach is equal o he larges processig ime amog he jobs i his bach. I his paper, we prove ha he problem 1 B V j is NP-hard. Keywords Schedulig; Bachig machie; Lae work; NP-hardess. 1 Iroducio The schedulig model ha we sudy is as follows. There are idepede jobs J 1,J 2,,J ha have o be scheduled o a ubouded bach machie. Each job J j ( j = 1,2,,) has a processig ime p j ad a due dae d j. All jobs are available for processig a ime 0. The goal is o schedule he jobs wihou preempio o he ubouded bach machie such ha he oal lae work is miimized. A bach machie is a machie ha ca process up o B jobs simulaeously. The jobs ha are processed ogeher form a bach. This model is moivaed by he problem of schedulig bur-i operaios for large-scale iegraed circui (IC) chips maufacurig (see Lee e al. [1] for deail). There are wo varias: he ubouded model, where B ; ad he bouded model, where b <. I his paper, we sudy he problem of schedulig idepede jobs o a ubouded bach machie o miimize he oal lae work. A schedule σ is a sequece of baches σ = (B 1,B 2,,B m ), where each bach B k (k = 1,,m) is a se of jobs ha are processed ogeher. The processig ime of bach B k is p(b k ) = max Jj B k {p j } ad is compleio ime is C(B k ) = k p(b q ). Noe ha q=1 Suppored by he Naioal Naural Sciece Foudaio (Gra Number ) ad he Provice Naural Sciece Foudaio of Shadog (Gra Number Y 2005A04). Correspodig auhor. qrjiafeg@163.com.

2 Miimizig he Toal Lae Work o a Ubouded Bach Machie 75 he compleio ime of job J j i σ, for each J j B k ad k = 1,,m, is C j (σ) = C(B k ). Whe here is o ambiguiy, we abbreviae C j (σ) o C j. The ardiess ad lae work of job J j are defied as T j = max{c j d j,0} ad V j = mi{t j, p j }, respecively. Usig he oaio of Graham e al. [2], we deoe his problem as 1 B Previous relaed work: The schedulig of bach processig machies is a impora research opic ad has araced a lo of aeio recely ([4,5]). I [6], Brucker e al. desig a dyamic programmig algorihm ha solves he problem of miimizig a arbirary regular cos fucio i pseudopolyomial ime. Whe he jobs have differe release imes, here has bee a lo of research work ([7,8]). As he objecive is o miimize he oal weighed lae work, Zhag ad Wag (2005) [3] prove ha 1 B NP-hard i he ordiary sese. Bu i is ope wheher 1 B solvable or biary NP-hard. V j. w j V j is V j is polyomially Our coribuio: I his paper, we prove he biary NP-hardess of 1 B This aswers he ope quesio posed i [3]. Our work s obaiig heavily depeds o he referece [9]. As he wo schedulig problems are differe models (such as he differe objecives eed differe aalysis) ad he problem 1 B very difficul by Zhag e al. [3]. I, herefore, is a differe work from [9]. 2 NP-hardess proof I his secio, we prove he NP-hardess of he problem 1 B from he NP-complee PARTITION problem. V j. V j has bee cosidered as V j by a reducio PARTITION. Give posiive iegers a 1,a 2,,a wih a i = 2B, decide if here exiss a pariio of he idex se {1,2,,} io wo disjoi subses X ad Y such ha a i = a i = B. i Y Give a isace of PARTITION, we firs defie iegers M = ( i)a i + 8B, M k = 2 i=k+1 L 1 = 7 L k = 2 k 1 ( i)a i + 8B, (k = 1,2,, 1) ( i)a i + 4B, L i + 7 ( i)a i + 4B, (k = 2,3,,2 + 1). I is easy o ge ha 2B < M < M 1 < < M 1 < L 1 < L 2 < < L 2+1. We defie a isace I of 1 B V j as follows. i X

3 76 The 7h Ieraioal Symposium o Operaios Research ad Is Applicaios I cosiss of jobs ha are classified io ypes. Each ype 2k 1 (1 k ) coais five jobs: J 1 2k 1,J2 2k 1,J3 2k 1 ad wo addiioal copies of J1 2k 1. Their processig imes ad due daes are give by p 1 2k 1 = L 2k 1, p 2 2k 1 = L 2k 1 + M k, d 1 2k 1 = 2 2k 2 d 2 2k 1 = 2 2k 1 L i + 5 k 1 L 2k 1 + M k + 2B, L i + 5 k 1 M i, p 3 2k 1 = L 2k 1 + 2M k, d2k 1 3 = 2 2k 1 L i + 5 2B. Each ype 2k (1 k ) also coais five jobs: J2k 1,J2 2k,J3 2k ad wo addiioal copies of J2k 1. Their processig imes ad due daes are give by p 1 2k = L 2k, d2k 1 = 2 2k 1 L i + 5 k 1 L 2k + 3M k + 2B, p 2 2k = L 2k + M k + a k, d 2 2k = 2 2k L i + 5 k 1 3M k ( k + 1)a k, p 3 2k = L 2k + 2M k, d2k 3 = 2 2k L i + 5 2B. Type coais hree copies of job J2+1 1 wih p = L 2+1, Se he hreshold value V = 2 d2+1 1 = L L i + 5 B. ( i)a i + B. Clearly, he cosrucio of he isace I of 1 B V j akes a polyomial ime uder he biary codig. Nex, we show ha he PARTITION isace has a soluio if ad oly if here exiss a schedule σ for he correspodig isace I wih V (σ) V, where V (σ) deoes he oal lae work of σ. Suppose ha I has a schedule σ = (B 1,B 2,,B m ) such ha V (σ) V, where each B i is a bach. The processig ime of bach B i is deoed by p(b i ). I is reasoable o require ha he processig ime of each job i B i+1 is larger ha p(b i ); oherwise, shifig he jobs i B i+1 wih processig imes o larger ha p(b i ) o B i does o icrease V (σ). The, we have he followig resul abou he schedule σ. Lemma 1. (1) The jobs i each B i come from a coiguous segme of he SPT sequece, ad all B i s are arraged i order of icreasig p(b i ); (2) For each k (1 k 2 + 1), all Jk 1 s are scheduled i a bach. Lemma 2. For he schedule σ. (1) Each bach coais oly jobs of oe ype; (2) For each k (1 k ), he jobs of ypes (2k 1) ad 2k are divided io four baches: {J2k 1 1,J2 2k 1 }, {J3 2k 1 }, {J1 2k }, {J2 2k,J3 2k } (paer oe), wih oal processig ime 2(L 2k 1 + L 2k ) + 5M k ; or {J2k 1 1 }, {J2 2k 1,J3 2k 1 }, {J1 2k,J2 2k }, {J3 2k } (paer wo), wih oal processig ime 2(L 2k 1 + L 2k) + 5M k + a k. Proof. If (1) does o hold, here mus exis some k (1 k 2 + 1) such ha Jk 3 ad J1 k+1 are scheduled i a bach. We cosider he ardiess of job Jk 3 s.

4 Miimizig he Toal Lae Work o a Ubouded Bach Machie 77 Firsly, we have ha Noe ha T 3 k p 1 k+1 d3 k = L k+1 d 3 k = (2 = 2 k > V. L i + 7 p 3 k 1 k > L k = Hece, he job Jk 3 s lae work is ( i)a i + 2B L i + 7 ( i)a i + 4B) (2 V 3 k = mi{p3 k,t 3 k } > V. k L i + 5 ( i)a i + 4B > V. 2B) A coradicio o V (σ) V, which implies ha each bach coais oly jobs of oe ype. Nex, we prove (2) by iducio. Firsly, we prove ha (2) is rue for k = 1 by provig he followig Observaio a 1 Observaio b 3. Observaio a 1. All jobs of ype 1 cao be scheduled i a bach. Proof. Suppose ha he jobs J1 1, J2 1 ad J3 1 are scheduled i a bach {J1 1,J2 1,J3 1 }. The, he oal ardiess of hree J1 1 s is 3T1 1 = 3(p 3 1 d1 1) = 3[L 1 + 2M 1 (L 1 + M 1 + 2B)] = 2M 1 + M 1 2B = V + B. Noe ha We obai p 1 1 = L 1 = 7 ( i)a i + 4B = V + 2B. V (J 1 1) = mi{ 1 3 [V + B], p 1 1} = 1 3 [V + B]. So he oal lae work of hree J1 1 s is 3V (J 1 1) = V + B > V. A coradicio o V (σ) V, which implies ha he jobs of ype 1 cao be scheduled i a bach. Observaio b 1. Jobs J1 1, J2 1 ad J3 1 cao be scheduled i hree baches. Proof. Suppose ha he jobs J1 1, J2 1 ad J3 1 are scheduled i hree baches {J1 1 }, {J2 1 } ad

5 78 The 7h Ieraioal Symposium o Operaios Research ad Is Applicaios {J1 3}. We ca easily ge ha he lae work of job J1 3 s is V (J 3 1 ) = mi{t 3 1, p3 1 } V + B. A coradicio o V (σ) V, which implies ha he jobs of ype 1 cao be scheduled i hree baches. From Observaio a 1 ad Observaio b 1, we ge ha jobs J1 1, J2 1 ad J3 1 mus be scheduled i wo baches: {J1 1,J2 1 }, {J3 1 }; or {J1 1 }, {J2 1,J3 1 }. Nex, we show ha he jobs of ype 2 also mus be scheduled i wo baches: {J2 1,J2 2 }, {J2 3}; or {J1 2 }, {J2 2,J3 2 }. Observaio a 2. All jobs of ype 2 cao be scheduled i a bach. Proof. Suppose ha he jobs J2 1, J2 2 ad J3 2 are scheduled i a bach {J1 2,J2 2,J3 2 }. The he oal lae work of hree J2 1 s is 3V (J 1 2) > V + B > V. A coradicio o V (σ) V, which implies ha he jobs of ype 2 cao be scheduled i a bach. Observaio b 2. Jobs J2 1, J2 2 ad J3 2 cao be scheduled i hree baches. Proof. Suppose ha he jobs J2 1, J2 2 ad J3 2 are scheduled i hree baches {J1 2 }, {J2 2 } ad {J2 3}. The, we also ca ge ha he lae work of job J2 3 s is V (J 3 2 ) = mi{t 3 2, p3 2 } V + B. A coradicio o V (σ) V, which implies ha he jobs of ype 1 cao be scheduled i hree baches. From Observaio a 2 ad Observaio b 2, we ge ha jobs J2 1, J2 2 ad J3 2 mus be scheduled i wo baches: {J2 1,J2 2 }, {J3 2 }; or {J1 2 }, {J2 2,J3 2 }. Observaio a 3. Baches {J1 1,J2 1 } ad {J1 2,J2 2 } are o exis i σ simulaeously. Proof. Suppose ha he baches {J1 1,J2 1 } ad {J1 2,J2 2 } are boh appear i he schedule σ. The he oal ardiess of hree J2 1 s is Noe ha So he oal lae work of hree J2 1 s is 3T2 1 = 3(p p p2 2 d2) 1 V + B. p 1 2 = L 2 > V + B. 3V (J 1 2) > V + B > V. A coradicio o V (σ) V. Observaio b 3. Baches {J1 2,J3 1 } ad {J2 2,J3 2 } are o exis i σ simulaeously. Proof. Suppose ha he baches {J1 2,J3 1 } ad {J2 2,J3 2 } are boh appear i he schedule σ. The he oal lae work of J1 2 ad J2 2 is V (J 2 1 J 2 2) = V (J 2 1) +V (J 2 2) = T T 2 2 > V.

6 Miimizig he Toal Lae Work o a Ubouded Bach Machie 79 A coradicio o V (σ) V. From he above Observaio a 1 Observaio b 3, he four baches of ype 1 ad ype 2 mus be {J 1 1,J 2 1},{J 3 1 },{J1 2},{J 2 2,J 3 2 };or{j1 1},{J 2 1,J 3 1 },{J1 2,J 2 2},{J 3 2 }. Tha is he coclusio (2) of Observaio 2 beig rue for k = 1. Secodly, we assume ha he coclusio (2) of Lemma 2 is rue for each i = 1,2, k 1. I is similar o prove ha he coclusio (2) of Lemma 2 is also rue for i = k, i.e., he jobs of ypes (2k 1) ad 2k mus be divided io he followig model four baches: {J2k 1 1,J2 2k 1 }, {J3 2k 1 }, {J1 2k }, {J2 2k,J3 2k } wih oal processig ime 2(L 2k 1 +L 2k )+5M k ; or {J2k 1 1 }, {J2 2k 1,J3 2k 1 }, {J1 2k,J2 2k }, {J3 2k }, wih oal processig ime 2(L 2k 1 + L 2k ) + 5M k + a k. Le X be he se of idices k (1 k ) such ha he four baches of ypes (2k 1) ad 2k are of paer oe. Le Y = Π \ X, where Π = {1,2,,}. A schedule wih properies of Lemma 1 ad Lemma 2 mus coai baches i he followig form. { {J 1 (B 4k 3,B 4k 2,B 4k 1,B 4k ) = 2k 1,J2k 1 2 },{J3 2k 1 },{J1 2k },{J2 2k,J3 2k }, k X, {J2k 1 1 },{J2 2k 1,J3 2k 1 },{J1 2k,J2 2k },{J3 2k }, k Y. B 4+1 = {J2+1 1 }. ( *) Lemma 3. For each k X, J2k 2 is he oly lae work job i B 4k 3, B 4k 2, B 4k 1 ad B 4k, ad is lae work is 2M k +( k +1)a k + a i. For each k Y, J2k 1 2 is he oly i<k, i Y lae work job i B 4k 3, B 4k 2, B 4k 1 ad B 4k, ad is lae work is 2M k + The lemma ca be proved by he mehod i [9]. Theorem 1. The problem 1 B V j is NP-hard. i<k, i Y Proof. The ime i akes o cosruc he schedulig isace is obviously polyomial. We show ha he PARTITION has a soluio if ad oly if here exiss a schedule σ for he schedulig isace I such ha V (σ) V. Firs, suppose ha X ad Y defie a soluio o PARTITION. Le σ be he schedule defied by (*). The schedulig σ has properies of Lemma 1, Lemma 2 ad Lemma 3. By Lemma 3 V (σ) = k=1 (2M k + i<k,i Y Where he hird erm is he oal lae work of hree J As So k=1 V (σ) = 2 i<k,i Y k=1 M k + a i ) + ( k + 1)a k + 3max{0, a k B}. k X a i = k=1 k=i+1 i<k,i Y a i = ( i)a i. i Y ( k)a k + 3max{0, a k B}. a i.

7 80 The 7h Ieraioal Symposium o Operaios Research ad Is Applicaios Noe ha By (2), we have V (σ) = 2 i X a i = a i = B. i Y ( i)a i + B = V. Coversely, suppose ha here exiss a schedule σ wih V (σ) V. From he proof of Lemma 1, Lemma 2 ad Lemma 3, he schedule σ mus have properies of Lemma 1, Lemma 2 ad Lemma 3. For he schedule σ, le X be he se of idices k (1 k ) such ha he four baches of ypes (2k 1) ad 2k are of paer oe ad Y = Π\X, where Π = {1,2,,}. By Noe ha we have From (1), we ge Applyig (1), (3) ad k X Y 3max{0, 3max{0, k X a k B} 3 a k 3B, a k B} 0. V (σ) V. a k B. (1) k X a k + 3 a k 3B B. (2) a k = 2B, we ge k X a k B. (3) a k = a k = B. Which shows ha X ad Y defie a soluio o PARTITION. Refereces [1] C.Y. Lee, R. Uzsoy ad L.A. Mari Vega, Efficie algorihms for schedulig semicoducor bur-i operaios, Operaios Research, , [2] R.L. Graham, E.L. Lawler, J.K. Lesra ad A.H.G. Riooy Ka, Opimizaio ad approximaio i deermiisic sequecig ad schedulig: a survey, Aals of Discree Mahemaics, , [3] Y. Zhag ad Wagli, The NP-compleeess of a ew bach schedulig problem, Joural of Sysem Sciece ad Mahmaic Sciece, , [4] C.N. Pos ad L.N. Va Wassehove, Sigle machie schedulig o miimize oal lae work, Operaios Research, ,

8 Miimizig he Toal Lae Work o a Ubouded Bach Machie 81 [5] C.N. Pos ad L.N. Va Wassehove, Approximaio algorihms for schedulig a sigle machie o miimize oal lae work, Operaios Research Leers, , [6] P. Brucker, A. Gladky, H. Hoogevee, M.Y. Kovalyov, C.N. Pos, T. Tauehah ad S.L. Va de Velde, Schedulig a bachig machie, Joural of Schedulig, , [7] Foo Afrai, Evripidis Bampis, Chadra Chekuri, David Karger, Claire Keyo, Sajeev Khaa, Ioais Milis, Maurice, Queyrae, Mari Skuella, Cliff sei, Maxim Svivideko, Approximaio schemes for miimizig average weighed compleio ime wih release daes, The Proceedig of he 40h Aual Symposium o Foudaios of Compuer Sciece, New York, 1999, [8] J. Re ad Y. Zhag, The PTAS for he problem of Pm r j,b C j, Aca Mahmaicae Applicae Siica, , [9] Z.H. Liu, J.J Yua ad T.C.E. Cheg, O schedulig a ubouded bach machie, Operaios Research Leers, ,