Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

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1 Absrac number: Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy 89 Nanjing hina) Absrac:This paper considers a single-machine scheduling wih prevenive periodic mainenance aciviies in remanufacuring sysem. Alhough he scheduling problem wih mainenance has araced researchers aenion mos of pas sudies consider only non-preempive jobs. In fac here exis boh preempive and non-preempive jobs due o heir differen processing echnic in remanufacuring workshop. Our objecive is o find a schedule o minimize he makespan and subjec o periodic mainenance and non-preempive jobs. In his paper an algorihm(we denoe i as -LS algorihm)is proposed. Is seps are as follows: ) Firsly scheduled he non-preempive jobs in machine by he (he Longes Processing Time)rule; )Then scheduled he preempive jobs ino he machine s remaining ime by he LS (he Lis Scheduling)rule. And more imporanly we discuss he wors-case raios of his algorihm in hree differen cases in erms of he value of he oal processing ime of he preempive jobs(we denoed i as S ): ) When S is longer han he remaining ime of he machine afer assigned he non-preempive jobs by he rule i is equal o ; ) When S falls in beween he remaining ime of he machine by he Biographies: Liu Biyu(98-) Female docoral suden [email protected]; hen Weida(965-) male docor professor [email protected].

2 rule and he rule i is less han ; 3) When S is less han he remaining ime of he machine by he rule i is less han. Key words: Single-machine scheduling; Prevenive periodic mainenance; Preempive jobs and Non-preempive jobs; -LS algorihm. Inroducion Mos lieraures on scheduling heory assume ha here are only non-preempive jobs. In fac his assumpion may no be valid in a real producion siuaion such as in he reprocessing workshop of remanufacuring indusry. There also exis preempive jobs. In remanufacuring indusry in order o reassemble finished producs new componens are required since he recovery rae of reurn componens can never reach 00%. In general a disassembly order is always released firs and hen he disassembly resul deermines wheher a purchasing order is needed []. And i is necessary o reprocess he disassembling cores as early as possible in order o deermine wheher purchase new componens or no. Our objecive is o find a schedule o minimize he makespan. For simpleness we only discuss off-line cores and assume ha he reprocessing ime have been deermined by workers experience. I is common o observe in pracice ha before machines are mainained i has some spare ime bu here is no a non-preempive job whose processing ime exacly is equal o he machine s spare ime and can be scheduled ino he machine in he same period. In order o making he bes of machine capaciy we can schedule he preempive jobs afer processing he non-preempive jobs ( In remanufacuring

3 indusry we assume here is zero seup ime). Therefore scheduling non-preempive and preempive jobs has gradually become a common pracice in many remanufacuring enerprise. Wih proper scheduling rule he workshop can improve producion efficiency and resuling in increased produciviy and high cusomers saisfacion degree []. Some lieraures have researched he single-machine scheduling problem wih single mainenance and non-preempive jobs. Liao and hen [3] have considered a scheduling problem for he objecive of minimizing he maximum ardiness. They propose a branch-and-bound algorihm o derive he opimal schedule and a heurisic algorihm for large-sized problems. They also provide compuaional resuls o demonsrae he efficiency of heir heurisic. Lee [4] shows ha he longes processing ime () rule has a igh wors-case raio of 4/3 for he objecive of minimizing he makespan and he also presens heurisics for oher objecives such as minimizing he maximum laeness he number of ardy jobs and he oal weighed compleion ime ec. Lee and Liman [5] prove ha he wors-case raio of he shores processing ime (SPT) rule is 9/7 for he objecive of minimizing he oal compleion ime. Graves and Lee [6] exend he problem o consider semi-preempive jobs. Ma e al. [7] discuss a single-machine scheduling problem wih an unavailable period in semipreempive case o minimize makespan and presen a -based heurisic. hen [8] considers a single-machine scheduling problem wih periodic mainenance o find a schedule ha minimizes he number of ardy jobs subjec o periodic mainenance and nonpreempive jobs. An effecive heurisic and a branch-and-bound algorihm are 3

4 proposed o find he opimal schedule. Wu and Lee [9] discuss a special problem where jobs processing ime is a decreasing funcion of is posiion in he schedule; when jobs are preempive i is shown ha he SPT rule provides he opimal schedule and for he nonpreempive case a mixed ineger programming echnique is proposed. Yang and Yang [0] sudied a single-machine scheduling wih he job-dependen aging effecs under muliple mainenance aciviies and variable mainenance duraion consideraions simulaneously o minimize he makespan. Ji e al. [] consider a single machine scheduling problem wih periodic mainenance for he objecive of minimizing he makespan. They show ha he wors-case raio of he classical Longes Processing Time firs algorihm is. In his paper we considered he scheduling problem wih periodic mainenance o minimize he makespan for boh non-preempive and preempive jobs. Firsly we proposed an algorihm by which he jobs are scheduled. Then we discussed he wors-case raios of his algorihm in hree differen cases in erms of he value of he oal processing ime of he preempive jobs.. Problem descripion and noaion Formally he considered problem could be described as follows: We assume here were n independen jobs J J J J J J J n n (Assume all jobs arrival ime is zero.) for reprocessing including non-preempive and preempive ones which were processed on a single machine. Here non-preempive means ha if a job can be finished before mainenance aciviy i has o resar; preempive means ha if a job can be finished before mainenance aciviy i can be 4

5 coninued o processed based on he previous process. The number of he former is n and he laer is n (Obviously n plus n is equal o n). They are defined group and group respecively. The processing ime of job J ij is p ij ( i ; when i hen j... n when i hen j... n ). We assumed ha T p j for every j... n for oherwise here is rivially no feasible schedule. Le ij be he compleion ime of job max J ij hen he objecive is o minimize he makespan which is defined as max i; j... ni () i j. Using he hree-field noaion of Graham e al. [] we denoed his scheduling problem as/ np p) pm / max. I can easily be shown ha his problem is srongly NP-hard [4] bu no approximaion algorihm has been provided and analyzed in he lieraure. We hough of each inerval beween wo consecuive mainenance aciviies as a bach wih a capaciy T. Thus a schedule can be denoed M L B L where M i is he ih mainenance aciviy L is he as B M B M... number of baches and B i is he ih bach of jobs. An illusraion of he considered problem in he form of a Gan char is given in Fig.. B B B L J [] J [j] J [i] M J [j+] M M L- J [n] J [n] T T T Fig.. An illusraion of he problem under consideraion where J [j] denoes he non-preempive job placed in he jh posiion of he given schedule. We use he wors-case raio o measure he qualiy of an approximaion algorihm. 5

6 Specifically for he makespan problem le P denoe he makespan produced by an approximaion algorihm P and denoe he makespan produced by an opimal off-line algorihm. Then he wors-case raio of algorihm P is defined as he smalles number c. (i.e. P c ) Some oher noaions are described as follows: T The lengh of ime inerval beween wo consecuive mainenance periods The amoun of ime o perform each mainenance aciviy The makespan(correspond o he load of bin packing problem) produced by an opimal off-line algorihm n The number of machine working periods(correspond o he bin numbers of bin packing problem) by an opimal off-line algorihm L The machine working ime s lengh in he las period by an opimal off-line algorihm( 0 L T ) The makespan produced by algorihm (i.e. he compleion ime of job J j ) n The number of machine working periods by algorihm L The machine working ime s lengh in he las period by algorihm ( 0 L T ) S The oal processing ime of Group jobs S The oal processing ime of Group jobs 3. The algorihm and is wors-case raio In his secion we analyze he algorihm (we define i as -LS algorihm) and 6

7 is wors-case raio. Before analyzing he -LS algorihm we firs define wo kinds of classical algorihms which are and LS algorihm respecively. The rule is a classical heurisic for solving scheduling problems. I can be formally described as follows. Algorihm. Re-order all he jobs such ha p jobs consecuively as early as possible. p p... n ; hen process he The LS algorihm is a classical polynomial ime approximaion algorihm for solving scheduling problems. I can be described as follows. Algorihm LS (Graham (966)). When a job is available (ies are broken arbirarily) assign i o he processing inerval of he machine where i can be finished as early as possible. Here we define -LS algorihm as follows: Algorihm -LS. The firs sep is o schedule he non-preempive jobs by he rule; he second sep is o schedule he preempive jobs by he LS rule for his considered scheduling problem. We le LS be he makespan of all being scheduled jobs(i.e. including non-preempive and preempive jobs). The makespan of opimal schedule is n T L () While he makespan of he schedule is n T L. () (3) subracs ()we obain 7

8 n n T L L (3) Lemma. (See Ji e al. (007))The wors-case raio of he algorihm for periodic mainenance is. (i.e.. ) This firs sep of he scheduling is similar o he problem ha Ji e al. (007) have researched. They prove ha he wors-case raio of he classical algorihm is. And hey show ha here is no polynomial ime approximaion algorihm wih a wors-case raio less han unless P = NP which implies ha he algorihm is he bes possible. So for non-preempive jobs he algorihm is also he bes possible. For he problem / np p) pm / max he wors-case raio of he LS algorihm exiss hree cases in erms of he value of S he conclusions are proved in he following heorems. Firsly we define a funcion: f ( x) k when k x k k is a posiive ineger. Si Obviously f ( ) is he frequency of machine mainenance where Si T is he oal processing ime of all being scheduled jobs. And i is easy o obain Si ni f ( ). T In he following we discuss he wors-case raio when S akes differen values. (4) Theorem 3.. When S S ( n ) LS. Proof. I means he oal processing ime of preempive jobs is longer han he machine spare ime afer scheduling non-preempive jobs. Obviously in his siuaion he preempive jobs can be assigned ino he machine s spare ime so ha he machine is always processing jobs unless mainenance. We can obain ha 8

9 S S LS S S f T ( ) ( ). (5) I is he opimal algorihm. Tha is LS. Insance. We assume here are 7 jobs in he reprocessing workshop. T n 7 n 3 n 4 p 6 p 8 p 5 p 3 So we can obain S n S ( n ) S S S ( n ) LS LS algorihm B B J J J = M J 3 J = 9 T T algorihm B B J J J = M J 3 J = 9 T T Fig.. An illusraion of he considered problem in case. nt Lemma. While n periods are used in he schedule S. Proof. Noe ha he oal processing ime of he jobs in any wo pairwise used bins is sricly larger han T by rule. And in all periods a mos one period in which he oal processing ime of he jobs are less han T/. Regardless of which period oal processing ime is less han T/ i plus any oher one period processing ime he resul mus larger han T. We discuss as follows: ase : If n is even hen 9

10 S n k B k n j p j bt (6) ase : If n is odd we can choose ( n ) bins including he bin in which he jobs processing ime is less han T/ and separae hem ino n groups wih wo pairwise. The oal processing ime of jobs in he remained bin (noe i B x )is larger han T/. Then S n k B k n j p j ( b -) T bt ( Bx) (7) Hence we obain Theorem 3.. When bt S S ( n ) S S ( n ) (8) LS. Proof. I means he oal processing ime of preempive jobs is shorer han he machine spare ime afer scheduling non-preempive jobs by rule and longer han he opimal algorihm. I is obviously ha nt ( n ) (9) Noe ha n S f ( ) T (0) ombining(8)(0)and Lemma 3 we can obain nt ( n ) nt ( n ) S S nt S S f ( ) S f ( ) f ( ) T T T nt ( n nt ( n ) ) () 0

11 Insance. We assume here are 8 jobs in he reprocessing workshop. T n 8 n 7 n p 4 p p 0 p p 6 p 4 p 3 p So we can obain S n 3 S ( n ) S ( n ) S.5 S ( n ) S S ( n ) LS B B B 3 -LS algorihm J J 4 M J J 5 M J 3 J 6 J 7 T T T algorihm B B B 3 J J 4 M J J 6 J 7 M J 3 J 5 T T T Fig.3. An illusraion of he considered problem in case. Theorem 3.3. When S S ( n ) LS. Proof. I means he oal processing ime of preempive jobs is shorer han he machine spare ime afer scheduling non-preempive jobs by he opimal algorihm. The preempive jobs can be scheduled ino he machine s spare ime. i.e. excluding he las bin he oher bins can be loaded full wihou any spare volume. So i s similar o he scheduling problem which Ji e al. (007) has researched. Therefore he wors-case raio of he -LS algorihm for he problem / nr r) pm / max in erms of he value of S )When S S ( n ) LS are as follows:

12 )When S ( n ) S S ( n ) LS 3)When S S ( n ) LS. 4. onclusions For he problem/ np p) pm / max his paper proposed a -LS algorihm and discussed he wors-case raios of his algorihm in hree differen cases in erms of he value of he oal processing ime of he preempive jobs which have been lised as Secion 3. In his research we assume he processing ime is fixed and off-line. In fuure research i is worh considering he processing ime is random and on-line. I is also worh considering he problem wih respec o oher objecives and in parallel-machine sysems. References [] Tang O Grubbsr O M R W Zanoni S. Planned lead ime deerminaion in a make-o-order remanufacuring sysem [J]. Inernaional Journal of Producion Economics (-): [] R.H.P.M. Ars Gerald M. Knapp Lawrence Mann Jr Some aspecs of measuring mainenance in he process indusry [J]. Journal of Qualiy in Mainenance Engineering 998 4():6. [3] Liao.J. hen W.J. Single-machine scheduling wih periodic mainenance and non-resumable jobs [J]. ompuers and Operaions Research (9): [4] Lee Y Machine scheduling wih an availabiliy consrain [J]. Journal of Global

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