Research Article Semi-Online Scheduling on Two Machines with GoS Levels and Partial Information of Processing Time

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1 e Scientific World Journal Volume 04, Article ID 57634, 6 pages Research Article Semi-Online Scheduling on Two Machines with GoS Levels and Partial Information of Processing Time Taibo Luo and Yinfeng Xu, Business School, Sichuan University, Chengdu 60065, China State Key Lab for Manufacturing Systems Engineering, Xi an 70049, China Correspondence should be addressed to Yinfeng Xu; yfxu@scu.edu.cn Received 5 November 03; Accepted 9 December 03; Published February 04 Academic Editors: J. G. Barbosa and D. Oron Copyright 04 T. Luo and Y. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates semi-online scheduling problems on two parallel machines under a grade of service (GoS)provision subject to minimize the makespan. We consider three different semi-online versions with knowing the total processing time of the jobs with higher GoS level, knowing the total processing time of the jobs with lower GoS level,orknowingbothinadvance.respectively,for the three semi-online versions, we develop algorithms with competitive ratios of 3/, 0/3, and 4/3 which are shown to be optimal.. Introduction Scheduling problem under a grade of service provision was first proposed by Hwang et al. []. The jobs should be assigned irrevocablytomachinesassoonastheyarrive.eachjob and machine are labeled with the GoS levels. A job can be processed by a particular machine if and only if the GoS level of the job is not less than that of the machine. In practice, the scheduling with GoS eligibility constraints is widely used. For more details, please refer to [ 3]. For the offline scheduling on m machines under a grade of service provision, Hwang et al. [] presented an lg-lpt algorithm which has a tight bound of 5/4 for two machines and /(m ) for m (m 3)machines.JiandCheng [4] gave an fptas for this problem. However, Woeginger [5] gave two simpler fptass for the same problem. For the online version, Jiang [6] proved a lower bound of for the case with two levels of GoS and presented an online algorithm with a competitive ratio of.5. Zhang et al. [7] improvedthe result and showed an algorithm with a competitive ratio of +(m m)/(m sm+s ) 7/3. For m =,Parketal.[8] presentedanoptimalalgorithm with a competitive ratio of 5/3. However,there are many papers focusing on the semi-online scheduling on two machines under GoS [8 3]. As the total processing time of all jobs is known, Park et al. [8] gave an optimal algorithm with a competitive ratio of 3/. When the largest processing timeofalljobsisknown,wuetal.[]presentedanoptimal algorithm with a competitive ratio of ( 5+)/.In the same paper, Wu et al. [] gave an optimal algorithm with a competitive ratio of 3/ whentheoptimalvalueoftheinstance is known. For the processing times bounded in an interval, Liu et al. [9] gave a competitive algorithm under some conditions and Zhang et al. [3] improved the result and gave an optimal algorithm for different intervals. In this paper, we focus on the semi-online scheduling problem where only partial information of the total processing time is known. The semi-online versions concerned in our paper are listed as follows. sum higher: the total processing time of all jobs with higher GoS level is known in advance. sum lower: the total processing time of all jobs with lower GoS level is known in advance. We use P GoS s C max to denote the semi-online problem with information s, where s {sum higher, sum lower}. Moreover,weuseP GoS s&s C max to denote the semi-online problem where both information s and informations are available in advance. Our results indicate that competitive ratios of three algorithms are better than 5/3 of the online version [8]. When knowing sum higher in advance, the competitive ratio is

2 The Scientific World Journal thesameastheoptimalalgorithm[8] presented for the semionline problem where the total processing time of all jobs is known. For designing algorithm, the results indicate that knowing sum higher&sum lower is more useful than sum higher or sum lower. Moreover,knowingsum higher is more useful than sum lower.however,knowingsum lower is better than knowing the largest processing time of all jobs []. The rest of this paper is organized as follows: in Section, we give some basic definitions. In Sections 3 5, we prove lower bounds and present algorithms for the three semionline problems, respectively.. Basic Definitions Wearegiventwomachinesandaseriesofjobsarrivingonline which are to be scheduled irrevocably at the time of their arrivals. The arrival of a new job occurs only after the current job is scheduled. We denote by the set J=,...,nthe set of all job indices arranged in the order of arrival. The jth arriving job is referred to as job J j. We denote each job by J j =(p j,g j ). The GoS assigned to job J j is denoted by g j,whichisifthe jobmustbeprocessedonlybythefirstmachineandifitmay beprocessedbyeitherofthetwo;p j is the processing time of job J j ; p j and g j arenotknownuntilthearrivalofjobj j.let T = gj = p j and T = gj = p j. The scheduled can be seen as the partition of J into two subsets, denoted by (S,S ),wheres and S contain job indicesassignedtothefirstandsecondmachine,respectively. Let t(s) = Σ Jj Sp j for an arbitrary subset S of J. Thenthe maximum of t(s ) and t(s ), denoted by max{t(s ), t(s )}, isthemakespanofthescheduled(s,s ).Theproblemisto minimize max{t(s ), t(s )}. The minimum makespan obtained by an optimal offline algorithm is denoted by. The makespan generated by algorithm A is denoted by C A. The competitive ratio of A is definedtobethemaximumc A /. 3. Optimal Algorithm for P GoS sum higher C max In this section, we prove a lower bound of competitive ratio and present an optimal algorithm for the semi-online version as T is known in advance. 3.. Lower Bound of Competitive Ratio Theorem. Any semi-online algorithm for P GoS sum higher C max has a competitive ratio of at least 3/. Proof. The theorem will be proved by adversary method. Let T =beknowninadvance.thefirstjobisj = (, ). J must be scheduled on the first machine. Then, we generate job J = (, ).IfjobJ = (, ) isscheduledonthefirstmachine, and there is no job arriving after, we have C A / =>3/. Otherwise, job J is scheduled on the second machine, and we generate job J 3 = (, ). No matter which machine that job J 3 is scheduled on, we have C A =3. Since the optimal algorithm will scheduled job J and job J on the first machine and scheduled job J 3 on the second machine. Hence, C A / = 3/.Theproofiscompleted. 3.. Optimal Semi-Online Algorithm GoS-TH. Sinceweknow T in advance and all the jobs with g j =must be scheduled on the first machine, we can regard them as one job; that is, J 0 =(T,). We scheduled job J 0 on the first machine at first and do not need to care about the job with g j =later. At the arrival of each job, H is updated to become a half of the total processing time of the jobs which include the jobs with GoS =and job J 0 ; P is updated to become the maximum processing time. We define P j, S j, Sj,andHj to be P, S, S,andH after we scheduled job J j. Then, clearly the optimum makespan L=max(T, P). Combinedwith the online algorithm presented by Park et al. [8], we propose Algorithm GoS-TH. Algorithm GoS-TH () Scheduled J 0 to the first machine and t(s 0 )=T. () Let p j =0for all the jobs with g j =. (3) Suppose that the incoming job is J j and g j =;assign job J j to the first machine. (4) Suppose the incoming job is J j and g j =. H j = H j +p j /, P j = max(p j,p j ),andl j = max(t j,p j ). (4.) If t(s j )+p j (3/)L j, scheduled it to the second machine. (4.) If t(s j )+p j > (3/)L j, scheduled it to the first machine. Theorem. The competitive ratio of Algorithm GoS-TH is 3/ for P GoS sum higher C max. Proof. Suppose that Theorem is false. There must exist an instance with the least number of jobs to make C GoS-TH > (3/). The makespan is not determined until the arrival of job J n. Therefore, C GoS-TH = max {t (S n ),t(sn )} > 3, max {t (S n ),t(s n )} 3. If g n =,basedonalgorithmgos-th, wehavet(s n )= t(s n ) and t(s n )=t(sn ). Due to the fact that max{t(s n ), t(s n )} (3/), we just need to prove Theorem is true when g n =. If g n =, when it is scheduled on the second machine, we have C GoS-TH =t(s n )=t(sn )+p n (3/).This is contradicting with inequality (). Otherwise, job J n is scheduled on the first machine, which leads to t(s n )+p n > (3/)L n (3/)H n ;combinedwith(t(s n )+p n +t(s n ))/ = H n,wegetthatt(s n ) < (/)L n (/).Sincep n,wehavet(s n )+p n < (3/),whichalsocontradicted with inequality (). The proof is completed. ()

3 The Scientific World Journal 3 4. Optimal Algorithm for P GoS sum lower C max In this section, we prove a lower bound of competitive ratio and present an optimal algorithm for the semi-online version as T is known in advance. 4.. Lower Bound of Competitive Ratio Theorem 3. Any semi-online algorithm for P GoS sum lower C max has a competitive ratio of at least 0/3. Proof. We will construct a job sequence with T =6to make an arbitrary algorithm A behave poorly. We begin with jobs J = (, ) and J = (, ). Wediscussthefollowingthree cases. Case (J and J are scheduled on the first machine). We continue to generate jobs J 3 = (, ) and J 4 = (, ). Ifjob J 3 and job J 4 are scheduled on the second machine, we have t(s )=4;thus,C A / = 4/4 > 0/3. Otherwise,if job J 3 or job J 4 isscheduledonthefirstmachineorbothof them are scheduled on the first machine, we further generate job J 5 = (6, ). Wewillhavet(S ) 40.Sinceoptimal algorithm will scheduled jobs J, J, J 3,andJ 4 on the second machine and scheduled job J 5 onthefirstmachine,wehave C A / 0/3. Case (J or J is scheduled on the first machine). We continue to generate job J 3 = (, ). Then we discuss the following two subcases. Subcase. (job J 3 is scheduled on the first machine). We continue to generate jobs J 4 = (, ) and J 5 = (, ). If job J 4 and job J 5 are scheduled on the second machine, we have t(s ) = 3 and t(s ) =, which lead to C A / = 3/3 > 0/3.Otherwise,ifjobJ 4 or job J 5 is scheduled on the first machine or both of them are scheduled on the first machine, we further generate job J 6 = (6, ) which leads to C A / 0/3. Subcase. (J 3 is scheduled on the second machine). We generate job J 4 = (5, ). IfjobJ 4 is scheduled on the second machine, we generate job J 5 = (3, ). IfjobJ 5 is scheduled on the second machine, we have t(s )=. Since the optimal algorithm will scheduled jobs J, J, J 3,andJ 4 on the first machine and scheduled job J 5 on the second machine, we have C A / = /3 > 0/3. IfjobJ 5 is scheduled on thefirstmachine,wegeneratejobj 6 = (6, ) which leads to C A / 0/3. Otherwise, if job J 4 is scheduled on the first machine, we generate job J 5 = (8, ). IfjobJ 5 is scheduled on the first machine, we generate job J 6 = (6, ) which leads to C A / 0/3. IfjobJ 5 is scheduled on the second machine, we generate job J 6 = (9, ).IfjobJ 6 is scheduled on thesecondmachine,wehavet(s )=0and =3;thus, C A / = 0/3. IfJ 6 is scheduled on the first machine, we generate job J 7 = (6, ) and have t(s )=40and =6, which also lead to C A / = 0/3. Case 3 (J and J are scheduled on the second machine). We continue to generate job J 3 = (, ). Then we discuss the following two subcases. Subcase 3. (job J 3 is scheduled on the second machine). We generate jobs J 4 = (3, ) and J 5 = (3, ). IfjobJ 4 or job J 5 is scheduled on the second machine or both of them are scheduled on the second machine, we further generate job J 6 = (, ). IfjobJ 6 is scheduled on the second machine, we generate job J 7 = (4, ). IfjobJ 7 is scheduled on the second machine, we have t(s ) =. Since the optimal algorithm will scheduled jobs J, J, J 3, J 4, J 5,andJ 6 on the first machine and scheduled job J 7 on the second machine, we have C A / /4 > 0/3. Otherwise,jobJ 7 is scheduled on the first machine, and then we generate job J 8 = (6, ) and have t(s ) 40 and = 6;hence, C A / > 0/3.IfjobJ 6 is scheduled on the first machine, we generate job J 7 = (3, ). IfjobJ 7 is scheduled on the second machine, we have t(s ) 0and =3,hence, we also have C A / > 0/3.Otherwise,jobJ 7 is scheduled on the first machine, and then we generate job J 8 = (6, ). We have C A / > 0/3. If job J 4 and job J 5 are scheduled on the first machine, we generate jobs J 6 = (8, ) and J 7 = (8, ). IfjobJ 6 or job J 7 is scheduled on the first machine or both of them are scheduled on the first machine, we further generate job J 8 = (6, ).We have C A / > 0/3. Otherwise,ifjobJ 6 and job J 7 are scheduled on the second machine, we have t(s )=0and =3, which lead to C A / = 0/3. Subcase 3. (job J 3 isscheduledonthefirstmachine).we generate job J 4 = (, ). IfjobJ 4 is scheduled on the first machine, we generate jobs J 5 = (0, ) and J 6 = (0, ). If job J 5 and job J 6 are scheduled on the second machine, we have t(s ) = and = 3;thus,C A / = /3 > 0/3. Otherwise,ifjobJ 5 or job J 6 is scheduled on the first machine or both of them are scheduled on the first machine, we further generate job J 7 = (6, ) and have t(s ) 40;thus, C A / = 0/3. Otherwise, if job J 4 is scheduled on the second machine, we generate job J 5 = (4, ).IfjobJ 5 is scheduled on the second machine, we generate job J 6 = (3, ). IfjobJ 6 is scheduled onthesecondmachine,wehavet(s )=and =3; hence, C A / = 0/3. IfjobJ 6 isscheduledonthefirst machine, we generate job J 7 = (6, ) and have t(s ) 4; thus, C A / = 4/6 > 0/3. Therefore, job J 5 is scheduled on the first machine. Then we generate jobs J 6 = (8, ) and J 7 = (8, ).IfjobJ 6 or job J 7 is scheduled on the first machine or both of them are scheduled on the first machine, we further generate job J 8 = (6, ).We will have t(s ) 40and =6;hence,C A / > 0/3. Otherwise, if job J 6 and job J 7 are scheduled on the second machine, we have t(s )=0and =3; also, we have C A / = 0/3.Theproofiscompleted. 4.. Optimal Semi-Online Algorithm GoS-TL. In this subsection, we design an optimal algorithm with a competitive ratio of 0/3. Letx and x be the jobs with g j = assigned

4 4 The Scientific World Journal to the first and second machine, respectively, where t(x )+ t(x ) = T. We define x j and xj to be x and x after we scheduled job J j.then,weproposealgorithmgos-tl. Algorithm GoS-TL () Suppose that the incoming job is J j and g j =;assign job J j to the first machine. () Suppose that the incoming job is J j and g j =. (.) (Stopping criterion ). If (3/3)T t(x j )+ p j (7/3)T, scheduled job J j and all the remaining jobs with GoS = to the first machine, and then scheduled all the remaining jobs with GoS = to the second machine. Stop. (.) (Stopping criterion ). If (6/3)T t(x j )+ p j (0/3)T, scheduled job J j to the second machineandscheduledalltheremainingjobsto the first machine. Stop. (.3) (Stopping criterion 3). If t(x j ) > 0/3,scheduled job J j and the remaining jobs to the first machine. Stop. (.4) If t(x j ) (7/6)T and (7/3)T t(x j )+ p j < (6/3)T, scheduled job J j to the first machine. Continue. (.5) Otherwise, scheduled job J j to the second machine. Continue. Lemma 4. If (6/3)T t(x ) (0/3)T,thenC GoS-TL (0/3). Proof. Since (/)T,wehavet(S ) = t(x ) (0/3)T (0/3). Then, we only need to prove that t(s ) (0/3).Wediscussitbythefollowingtwocases. Case (T T ). In this case, we have t(s )=t(x )+T and T (T +T )/. Since(6/3)T t(x ) (0/3)T,we have t(x ) (7/3)T.Combinedwith T,wehave C GoS-TL (7/3) T +T T (7/3) T +T T = 0 3. () Case (T <T ). In this case, we have T <(T +T )/,which leads to (T +T )/.Then,wehave C GoS-TL (4/3) T +T T +T. (3) Since ((4/3)T +T )/(T +T ) is increasing function of the variation of T,thus,wehave C GoS-TL (4/3) T +T T +T < (4/3) T +T T +T = 0 3. (4) The proof is completed. BasedonLemma4, we straightforwardly have Corollary 5. Corollary 5. () If t(x ) (7/3)T,thent(S ) (0/3) ; () if t(x ) (0/3)T,thent(S ) (0/3). Lemma 6. t(x ) (7/3)T. Proof. Since Algorithm GoS-TL will only scheduled a job with g j =to the first machine only when t(x j )+p j (7/3)T, the lemma can directly be got from the Algorithm GoS-TL.Theproofiscompleted. By using Corollary 5 and Lemma 6, we can obtain the following corollary. Corollary 7. If t(x ) (0/3)T, then C GoS-TL (0/3). BasedonLemma6 and Corollary 7, ifweprovet(s ) (0/3) will hold when t(x ) > (0/3)T,thenwecan prove that Algorithm GoS-TL is (0/3)-competitive. Lemma 8. If job J j is scheduled on the second machine by Algorithm GoS-TL where t(x j )+p j > (0/3)T and t(x j ) (7/6)T,thent(S ) < (0/3). Proof. If t(x j )+p j > (0/3)T and t(x j ) (7/3)T,we have p j > (/)T.Thenwehave p j > (/)T.IfjobJ j is scheduled on the second machine, Algorithm GoS-TL will scheduled the remaining jobs on the first machine. Thus, t(s j ) t(s j ) p j = t(x j )+p j. (5) p j Since (t(x j )+p j )/p j is decreasing function of the variation of p j and increasing function of variation of t(x j ),wehave t(s j ) t(x j )+p j < (7/6) T + (/) T = 0 p j (/) T 3. (6) The proof is completed. Lemma 9. If (6/3)T t(x j ) (0/3)T,thent(S )> (0/3)T will never happen. Proof. If (6/3)T t(x j ) (0/3)T, Algorithm GoS-TL will scheduled the remaining jobs on the first machine, so t(s )=t(x j ) (0/3)T.Theproofiscompleted. Lemma 0. If job J j is scheduled on the second machine by Algorithm GoS-TL where (7/6)T <t(x j ) < (6/3)T and t(x j ) > (0/3)T,thent(S )=t(x j ) < (0/3). Proof. Let J w be the job that is scheduled on the second machine by Algorithm GoS-TL which satisfies t(x w ) 7/3 and (7/6)T <t(x w )+p w < (6/3)T.SinceJ w is

5 The Scientific World Journal 5 scheduled on the second machine by Algorithm GoS-TL and (7/6)T <t(x w )+p w < (6/3)T,wehavet(x w )+p w > (7/3)T ; otherwise, Algorithm GoS-TL will scheduled job J w on the first machine. Moreover, t(x w ) < (3/3)T must hold which leads to p w > (4/3)T.Thisfurtherimpliesthat t(x w ) < (/3)T since t(x w )+p w < (6/3)T. If t(x w ) = 0,thenp w > (7/3)T which leads to t(x w ) > (6/3)T ; this is contradicting with the definition of job J w. Therefore, we have t(x w ) > 0.Basedonstep (.4) of Algorithm GoS-TL,theprocessingtimeof thejobthat is assigned to x w is larger than (3/6)T since t(x w ) < (/3)T.Combinedwitht(x w ) < (3/3)T,weknowthere is only one job in x w,callitjobj V.SincejobJ V,jobJ w,and t(x w ) need to satisfy p V +p w > 7 3 T, p V +t(x w )> 7 6 T, 7 6 T <t(x w )+p w < 6 3 T, we can get p V > (9/5)T, which implies that t(s ) < (43/5)T. Since there is no job with g j = that will scheduled on the first machine between job J V and J j by Algorithm GoS-TL,combinedwithjobJ j being scheduled on the second machine, we have p V +p j > (7/3)T.Sincep j > (4/3)T, p V, p w,andp j must satisfy p V +p w > 7 3 T, p V +p j > 7 3 T, p w +p j 9 3 T, which implies that > (7/3)T,hence, The proof is completed. (7) (8) t(s ) < (43/5) T (7/3) T = 0 3. (9) Based on Lemma 4 to Lemma 0 and Corollary 5 to Corollary7, we have the following theorem naturally. Theorem. The competitive ratio of Algorithm GoS-TL is 0/3 for P GoS sum lower C max. 5. Optimal Algorithm for P GoS sum high&sum lower C max In this section, we show a lower bound of competitive ratio and present an optimal algorithm for the semi-online version as T and T are known in advance. 5.. Lower Bound of Competitive Ratio Theorem. Any semi-online algorithm for P GoS sum high&sum lower C max has a competitive ratio of at least 4/3. Proof. The theorem will be proved by adversary method. Let T = /3 and T = 5/3 be known in advance. The first job is J = (/3, ). JobJ must be scheduled on the first machine. Then job J = (/3, ) arrives. If job J is scheduled on the second machine, we further generate jobs J 3 = (, ) and J 4 = (/3, ). Inthissituation,wehaveC A 4/3and =since the optimal algorithm will scheduled jobs J, J,andJ 4 onthefirstmachineandscheduledjobj 3 on the second machine. Thus, we have C A / 4/3.Otherwise, if job J is scheduled on the first machine, then we further generate jobs J 3 = (/3, ) and J 4 = (/3, ).Inthissituation, we also have C A 4/3 and = since the optimal algorithm will scheduled jobs J and J 3 on the first machine and scheduled jobs J and J 3 on the second machine. The proof is completed. 5.. Optimal Semi-Online Algorithm GoS-TB. In this subsection, we design an optimal algorithm with a competitive ratio of 4/3. SinceweknowT in advance and all the jobs with g j =must be scheduled on the first machine, therefore, we can regard them as one job, that is, J 0 =(T,). We scheduled job J 0 onthefirstmachineatfirstanddonotneedtocare about the job with g j =later. We present Algorithm GoS-TB as follows. Algorithm GoS-TB () Scheduled J 0 to the first machine and t(s 0 )=T. () Let p j =0for all the jobs with g j =. (3) Suppose that the incoming job is J j and g j =;assign job J j to the first machine. (4) Suppose that the incoming job is J j and g j =. (4.) If t(s )+p j (/3)(T +T ), scheduled it to the first machine. (4.) (Stopping criterion). Suppose t(s ) + p j > (/3)(T +T ). (4..) If t(s j ) > (/3)(T +T ), scheduled job J j and the remaining jobs with GoS = on the second machine and scheduled the remaining jobs with GoS = to the first machine. Stop. (4..) If t(s j ) (/3)(T +T ), scheduled job J j to the second machine, and scheduled the remaining jobs to the first machine. Stop. Theorem 3. The competitive ratio of Algorithm GoS-TH is 4/3 for P GoS sum high&sum lower C max. Proof. Since the job with g j =isscheduledatfirstandwe donotneedtocareaboutthemafterthat.wefocusonthe jobs with g j =.

6 6 The Scientific World Journal Assume job J j is the first job with g j =to make t(s )+ p j > (/3)(T +T ). We prove it by the following two cases. Case (t(s j ) > (/3)(T +T )). In this case, Algorithm GoS-TB schedules job J j and the remaining jobs with GoS = on the second machine. Therefore, t(s ) (/3)(T +T ) and t(s ) T +T t(s ) < (/3)(T +T ).Since (/)(T + T ),wehavec GoS-TH = max{t(s ), t(s )} (/3)(T +T ) (4/3). Case (t(s j ) (/3)(T +T )). In this case, we have p j > (/3)(T +T ). Algorithm GoS-TH will only scheduled job J j onthesecondmachineandscheduledtheremainingjobson the first machine. If p j (/)(T +T ),wehave =p j and t(s ) p j =t(s ). Therefore, C GoS-TH =.Otherwise, (/3)(T +T )<p j < (/)(T +T ), which leads to t(s ) T +T t(s ) < (/3)(T +T ).Hence,wehaveC GoS-TH = max{t(s ), t(s )} (/3)(T +T ) (4/3).Theproofis completed. [9] M.Liu,C.Chu,Y.Xu,andF.Zheng, Semi-onlinescheduling on machines under a grade of service provision with bounded processing times, Combinatorial Optimization, vol.,no.,pp.38 49,0. [0] M. Liu, Y. Xu, C. Chu, and F. Zheng, Online scheduling on two uniform machines to minimize the makespan, Theoretical Computer Science,vol.40,no.-3,pp ,009. [] X. Lu and Z. Liu, Semi-online scheduling problems on two uniform machines under a grade of service provision, Theoretical Computer Science, vol , pp , 03. [] Y. Wu, M. Ji, and Q. Yang, Optimal semi-online scheduling algorithms on two parallel identical machines under a grade of service provision, International Production Economics,vol.35,no.,pp ,0. [3] A.Zhang,Y.Jiang,L.Fan,andJ.Hu, Optimalonlinealgorithms on two hierarchical machines with tighly-grouped processing times, Combinatorial Optimization,03. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments ThisworkissupportedbyNationalNaturalScienceFoundation of China under Grants 7073, , and 7379 and Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT73. References [] H.-C. Hwang, S. Y. Chang, and K. Lee, Parallel machine scheduling under a grade of service provision, Computers and Operations Research,vol.3,no.,pp ,004. [] A.Bar-Noy,A.Freund,andJ.Naor, On-lineloadbalancingina hierarchical server topology, SIAM Journal on Computing,vol. 3, no., pp , 00. [3] J. Ou, J. Y.-T. Leung, and C.-L. Li, Scheduling parallel machines with inclusive processing set restrictions, Naval Research Logistics,vol.55,no.4,pp ,008. [4]M.JiandT.C.E.Cheng, AnFPTASforparallel-machine scheduling under a grade of service provision to minimize makespan, Information Processing Letters, vol. 08, no. 4, pp. 7 74, 008. [5] G. J. Woeginger, A comment on parallel-machine scheduling under a grade of service provision to minimize makespan, Information Processing Letters,vol.09,no.7,pp.34 34,009. [6] Y.Jiang, OnlineschedulingonparallelmachineswithtwoGoS levels, Combinatorial Optimization, vol. 6, no., pp. 8 38, 008. [7] A.Zhang,Y.Jiang,andZ.Tan, Onlineparallelmachinesscheduling with two hierarchies, Theoretical Computer Science, vol. 40, no , pp , 009. [8] J. Park, S. Y. Chang, and K. Lee, Online and semi-online scheduling of two machines under a grade of service provision, Operations Research Letters,vol.34,no.6,pp ,006.

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