Online Over Time Scheduling on Parallel-Batch Machines: A Survey

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1 J. Oer. Res. Soc. China (014) :44 44 DOI /s Online Over Time Scheduling on Parallel-Batch Machines: A Survey Ji Tian Ruyan Fu Jinjiang Yuan Received: 30 October 014 / Revised: 3 November 014 / Acceted: 4 November 014 / Published online: 6 November 014 Ó Oerations Research Society of China, Periodicals Agency of Shanghai University, and Sringer- Verlag Berlin Heidelberg 014 Abstract Online scheduling is a raidly develoed branch in scheduling theory. In this aer, we resent an extensive survey for online over time scheduling on arallel-batch machines. Some oen roblems are roosed for further research. Keywords Online scheduling Parallel-batch machines Cometitive ratio 1 Introduction In arallel-batch scheduling, we have n jobs J 1 ; ; J n and m arallel-batch machines M 1 ; ; M m. Each job J j has a release date r j > 0, a rocessing time j [ 0, a weight w j [ 0, and a delivery time q j > 0. Each arallel-batch machine M i can rocess u to b jobs simultaneously as a batch. Here, a batch is a subset of jobs and b is the caacity of the batches. If b ¼1, the model is called unbounded batching. Alternatively, if b\1, the model is called bounded batching. The rocessing time of a batch is defined as the maximum rocessing time of the jobs in the batch. A batch can be started to rocessing at a time t if all the jobs in the batch are released by time t. Furthermore, in a schedule, the jobs in a batch have the same starting time and the same comletion time, resectively. Then a schedule for This work was suorted by National Natural Science Foundation of China (NSFC, No ) and NSF of Jiangsu Province (No. BK ). Fu was also suorted by NSFC (No ), and Yuan was also suorted by NSFC (Nos and ). J. Tian R. Fu College of Sciences, China University of Mining and Technology, Xuzhou 1116, Jiangsu, China J. Yuan (&) School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 40001, China yuanjj@zzu.edu.cn 13

2 446 J. Tian et al. the arallel-batch scheduling can be determined by artitioning the jobs into batches and assigning the batches to the available time saces of the m arallel-batch machines without overla. Parallel-batch scheduling was motivated by semiconductor manufacturing. Uzsoy et al. [4, 46], Avramidis et al. [], and Mathirajan and Sivakumar [31] described the alication in the semiconductor manufacturing rocess in detail. The fundamental model of bounded arallel-batch scheduling was first introduced by Lee et al. [1]. For roblem 1j batch; b\1jc max, it was reorted by Lee and Uzsoy [] that the otimal schedule is given by the full batch longest rocessing time (FBLPT) rule roosed first by Bartholdi. An extensive discussion of the unbounded arallelbatch scheduling roblem was rovided by Brucker et al. [4]. Recent develoments on this toic can be found in [] and [6]. With dynamic job arrivals and the caacity b being infinite, Lee and Uzsoy [] resented a dynamic rogramming algorithm to solve roblem 1j batch; b ¼1; r j jc max in Oðn Þ time. For the same roblem, Poon and Zhang [34] resented an imroved Oðn log nþ-time algorithm. For the bounded arallel-batch scheduling roblem 1j batch; b\1; r j jc max, Liu and Yu [6] showed that the roblem with only two arrival times is NP-hard, and gave a seudo-olynomial-time algorithm in case of fixed number of arrival times. Brucker et al. [4] roved that the general roblem is NP-hard in the strong sense. Online scheduling is a relatively new toic in scheduling research and has been extensively studied in the last decade. While there are different meanings of online scheduling, the term online in this aer means that jobs arrive over time. In the online over time environment, jobs arrive over time and we do not have any information about the jobs in advance. The information of each job J j can be known only at the arrival time r j of the job. Hence, we must schedule the available jobs without information of the future jobs. Usually, the information of a job J j includes its release time r j, its rocessing time j, its delivery time q j, its weight w j, its job family, etc. In the makesan minimization, the necessary information of a job is its release time r j, its rocessing time j, and its job family (when the jobs are artitioned into incomatible job families). The quality of an online algorithm is measured by its cometitive ratio. Suose that we are considering an online scheduling roblem P to minimize a certain objective function. Let C on ðlþ and C ot ðlþ denote, resectively, the objective value of an online algorithm H and the off-line otimal value for an inut L. We say that algorithm H is R-cometitive for some R > 1ifC on ðlþ=c ot ðlþ 6 R for all inuts L. The cometitive ratio R H of algorithm H is defined as R H ¼ su fc on ðlþ=c ot ðlþg: L In this case, we also say that algorithm H is R H -cometitive, and R H is an uer bound of the cometitive ratio of roblem P. Given an online scheduling roblem P, we say that it has a lower bound q, if every online algorithm for P has a cometitive ratio at least q. Furthermore, if A is an online algorithm for P such that the cometitive ratio of A is exactly the lower bound q for P, we say that A is the best ossible online algorithm for P. 13

3 Online Over Time Scheduling on Parallel-Batch Machines 447 Some examles of the studies on online scheduling roblems (with jobs arriving over time) can be found in [1, 9, 18, 19], and [1], among many others. In general, the cometitive ratio of an online algorithm may be imroved if some information of the jobs is known in advance. This scenario is described as semionline in the literature. There have been lenty researches concerning semi-online scheduling with jobs arriving over list. For examle, Cheng et al. [8] studied the semi-online scheduling on arallel machines with given total rocessing time. Seiden et al. [37] studied the semi-online scheduling on arallel machines with decreasing job sizes. Tan and He [38] studied semi-online scheduling on two arallel machines with combined artial information. In contrast, there are only a few researches concerning semi-online scheduling with jobs arriving over time. The reresentative ublication was given by Hall et al. [17]. They studied the semionline scheduling on a single machine to minimize the sum of weighted comletion time with known arrival times of the jobs. In this survey, we reort the develoments on online scheduling of arallel-batch machines. An informal version of this aer [47] was resented in the web site htt:// Let f be the objective function to be minimized. Using the standard scheduling classification scheme of Lawler et al. [0], the scheduling models considered in this survey can be summarized as Pjonline; -batch; b; bjf ; where b \ n or b ¼1, b frestart; L-restart; f families; Agreeableð j ; q j Þ; q j > j ; q j 6 j ; J ðtþ; ðtþ; r ðtþg and f fc max ; D max ; F max ; P C j ; P w j C j g. The notations aearing in the above models are described as follows: C j is the comletion time of job J j. C max ¼ maxfc j : 1 6 j 6 ng is the maximum comletion time of all jobs. D max ¼ maxfc j þ q j : 1 6 j 6 ng is the maximum delivery time by which all jobs have been delivered. F max ¼ maxfc j r j : 1 6 j 6 ng is the maximum flow time of all jobs. P 16j6n ðw jþc j is the total (weighted) comletion time of jobs. Restart means that a running task may be interruted, losing all the work done on it. In this case, an interruted job becomes unscheduled. Allowing restart reduces the imact of a wrong decision. L-restart means that batches are only allowed limited restarts. If a batch has been restarted one time, then all jobs in it are considered as interruted jobs. Any new batch that contains interruted jobs cannot be restarted any more. That is, any job is allowed to restart only once. f families means that the jobs are artitioned into f incomatible families so that the jobs in different families cannot be rocessed in a common batch. Agreeableð j ; q j Þ means that for jobs J i and J j,if i > j, then q i > q j. q j 6 j means that, for each job J j, the rocessing time j is no less than the delivery time q j. 13

4 448 J. Tian et al. q j > j means that, for each job J j, the rocessing time is no longer than the delivery time q j. J ðtþ means that at time t the information of the first longest job arriving after time t is given. ðtþ means that at time t the rocessing time of the first longest job arriving after time t is given. r ðtþ means that at time t the arrival time of the first longest job arriving after time t is given. To Minimize Makesan C max Many researchers focused on studying online scheduling on arallel-batch machines to minimize makesan, and they rovided so many distinct online algorithms, even for the same roblem. In fact, online algorithms could be divided into two classes. One is called delay algorithm, which means that the algorithm always waits until the time moment satisfying some conditions, even the jobs are available and some machines are idle. The other one is called greedy algorithm which always immediately assigns available jobs as long as there exist idle machines, with no waiting time. Zhang et al. [1] and Deng et al. [9] resented basic ideas for the research of online scheduling on arallel-batch machines, which have been widely acceted in the onward research. They investigated online scheduling on a single batch machine to minimize makesan. For unbounded version, they indeendently rovided the same best ossible ffiffi online delay algorithm with a cometitive ratio of þ1. Later, Poon and Yu [3] further rovided a best flexible arameterized online algorithm, which contains the above algorithm. For the bounded case, Lee and Uzsoy [] resented a greedy algorithm, which was roved to be -cometitive by Liu and Yu [6]. Zhang et al. [1] roosed two algorithms based on FBLPT rule with cometitive ratio, while they roved that no online algorithm has a cometitive ratio less than ffiffi. Poon and Yu [36] resented a class of FBLPT (full batch longest rocessing time)-based algorithms including the above three algorithms with cometitive ratio. For the secial case b ¼, they rovided an online algorithm with a cometitive ratio of 7/4. Fu et al. [14] investigated the unbounded model with restarts. They roved that no online algorithm has a cometitive ratio less than ffiffi, and they resented a 3 - cometitive algorithm. Later, Yuan et al. [49] designed a best ossible online algorithm with a cometitive ratio ffiffi. Fu et al. [1] studied the same roblem with limited restarts, in which a job cannot be restarted twice. They roved that the best online algorithm is 3 -cometitive. With restarts, these results imrove the known ffiffi -cometitive. For the bounded model with restarts or limited restarts, Liu and Yuan [9] studied the secial case that all jobs have equal rocessing time. They roosed two best ossible algorithms with four different cometitive ratios deending on the caacity b of a batch. There are several results about the online scheduling roblem on a single batch machine with job families but only the unbounded model was investigated. When 13

5 Online Over Time Scheduling on Parallel-Batch Machines 449 the number of families is, Fu et al. [16] roosed a best ossible online algorithm with cometitive ratio ffiffiffi 17 þ3 4. For the general version with f families, Nong et al. [3] firstly rovided an online algorithm with cometitive ratio and roved that the algorithm is best ossible when f tends to infinity. Lastly, when the number f of job families is given in advance, Fu et al. [13] rovided a best ossible online ffiffiffiffiffiffiffiffiffi 4f algorithm with a cometitive ratio of 1 þ þ1 1 f, which extends the secial case f ¼ 1 including the results for unbounded case in Zhang et al. [1], Deng et al. [9], and Poon and Yu [3]. For semi-online scheduling on a single unbounded batch machine to minimize makesan, given the information J ðtþ (the first longest job) or ðtþ (the rocessing time of the first longest job) arriving after time t, Yuan et al. [0] roosed best ossible online algorithms with cometitive ratio ffiffi. The result imlies that the information ðtþ dominates r ðtþ since the information J ðtþ consists of ðtþ and r ðtþ which is the arrival time of the first longest job arriving after time t. Given the information r ðtþ, Yuan et al. [0] rovided a 3 -cometitive algorithm and showed a lower bound Online scheduling on m arallel-batch machines was extensively studied as well. ffiffi For the unbounded version, Zhang et al. [1] rovided a lower bound mþ1 of cometitive ratio and resented an online algorithm PHðh m Þ with a cometitive ratio of 1 þ h m, where 0\h m \1 and h m ¼ð1 h m Þ m 1. In the case that jobs have identical rocessing time, they roosed two best ossible online algorithms with cometitive ratios of 1 þ b m, where ð1 þ bþ mþ1 ¼ b þ, for b ¼1and ffiffi for b\1. When m ¼, Nong et al. [33] resented an online algorithm with ffiffi cometitive ratio. Later, Tian et al. [41] roved that the lower bound of this ffiffi roblem is and they designed a distinct best ossible algorithm. For the general version with m being arbitrary, Tian et al. [40] and Liu et al. [7] indeendently resented two distinct best ossible algorithms with a cometitive ratio of 1 þ c m, where c m ¼ ffiffiffiffiffiffiffiffi m þ4 m : Tian et al. [40] also rovided a best ossible 3 -cometitive dense algorithm that always immediately rocesses the available jobs as a batch on one idle machine when there are at least two idle machines. Their result generalized the work of Nong et al. [33] and Tian et al. [41] for m ¼. Tian et al. [4] also considered the same roblem with incomatible families, where the number of families is equal to the number of machines m. They first gave a lower bound ffiffi on the cometitive ratio of any online algorithm, then they rovided an online algorithm H m ðhþ with a arameter 0\h\1, and showed that its cometitive ratio is no less than 1 þ ffiffiffi 10 1:63. For the case f ¼ and h ¼ ffiffi 1, the authors roved that the algorithm H m ðhþ is best ossible with cometitive ratio ffiffi. When m > 3 and h ¼ ffiffiffi 1, the cometitive ratio of algorithm Hm ðhþ is no larger than 1 þ ffiffi 1:707. A class of above results is listed in Table 1. In the table, if the lower bound is equal to the uer bound of the same roblem, we say the roblem has a best 13

6 40 J. Tian et al. Table 1 For minimizing C max Problem Lower bound Uer bound References 1jonline; -batch; b ¼1jC max ffiffi ffiffi þ1 [9, 3, 1] 1jonline, -batch, b\1ðor b ¼ ÞjC max ffiffi þ1 (or 1.7) [36, 1] 1jonline; -batch; b ¼1; restartjc max ffiffi ffiffi [14, 49] 1jonline; -batch; b ¼1; L-restartjC max 1: 1: [1] 1jonline; -batch; j ¼ ; b ¼ ; L-restartjC max 1 þ a 1 þ a [9] 1jonline; -batch; j ¼ ; b > 3; L-restartjC max 1 þ b 1 þ b [9] 1jonline; -batch; j ¼ ; b ¼ ; restartjc max 1 þ a 1 þ a [9] 1jonline; -batch; j ¼ ; b ¼ 3; restartjc max 1 þ c 1 þ c [9] 1jonline; -batch; j ¼ ; b > 4; restartjc max 1 þ u 1 þ u [9] ffiffiffi ffiffiffi 1jonline; -batch; b ¼1; familiesjc max 17 þ3 17 þ3 [16] 4 4 1jonline; -batch; b ¼1; f families jc max 1 þ a f 1 þ a f [13] 1jonline; -batch; b ¼1; J ðtþðor ðtþþjc max ffiffi ffiffi [0] 1jonline; -batch; b ¼1; r ðtþjc max 1:44 1: [0] ffiffiffi ffiffiffi Pjonline; -batch; b ¼1jC max [33, 41] ffiffi Pjonline; -batch; L-restart; b ¼1jC max 1:98 3 þ1 [1] Pmjonline; -batch; j ¼ ; b ¼1jC max 1 þ b m 1 þ b m [] ffiffi ffiffi Pmjonline; -batch; j ¼ ; b\1jc max þ1 [] Pmjonline; -batch; b ¼1jC max 1 þ c m 1 þ c m [7, 40] ffiffi Pmjonline; -batch; m families; b ¼1jC max þ1 1 þ ffiffi [4] ossible online algorithm and it has been solved thoroughly. It leaves the status of the roblem oen if there exists a ga between the lower bound and the uer bound. Two additional notations will aear in the table. Let b m be the ositive solution of equation ð1 þ bþ mþ1 ¼ b þ. Let a f be the ositive solution of equation ffiffiffiffiffiffiffiffiffi f a 4f þ a f ¼ 0, i.e., a f ¼ 1 þ þ1 1 f. Let c m be the ositive solution of equation c þ mc 1 ¼ 0, i.e., c m ¼ ffiffiffiffiffiffiffiffi m þ4 m : Let a 0:9; b 0:47; c 0:38, and u 0:40 be the ositive solution of ð1 þ aþða þ 4a þ 1Þ ¼3, bð1 þ bþ ¼ 1, cðc þ 1Þðc þ 3Þ ¼, and uðu þ 1Þðu þ 1Þ ¼1, resectively. We resent the following oen roblems for further research: (1) Pmjonline; -batch; f families; b ¼1jC max. () Pmjonline; -batch; restart; b ¼1jC max. (3) Pmjonline; -batch; L-restart; b ¼1jC max. (4) Pmjonline; -batch; J ðtþ; b ¼1jC max. () Pmjonline; -batch; ðtþ; b ¼1jC max. (6) Pmjonline; -batch; r ðtþ; b ¼1jC max. Furthermore, for roblem 1jonline; -batch; b\1jc max, the known lower bound is and the known uer bound is. Poon and Yu [36] resented an online 13

7 Online Over Time Scheduling on Parallel-Batch Machines 41 algorithm of cometitive ratio 7=4 when b ¼. For general b, it remains a long standing and challenging oen roblem. 3 To Minimize the Maximum Delivery Time D max Many researchers studied online scheduling roblems with delivery times. The objective is to minimize the time by which all jobs have been delivered, denoted by D max. Hoogeveen and Vestjens [19] first studied online roblem on a single machine with delivery time, and they resented a best ossible online algorithm with cometitive ratio ffiffi : Recently, Tian et al. first investigated an online model with delivery time on a arallel-batch machine. Tian et al. [39] roosed three online algorithms to solve the roblem on a single batch machine to minimize the maximum delivery time. The lower bound ffiffi resented by Zhang et al. [1] is also aroriate for this model. Tian et al. [39] rovided a -cometitive algorithm for b ¼1 and a 3-cometitive algorithm for b\1. When the jobs have equal rocessing time, they resent best ossible algorithms for arbitrary batch caacity b. Under unbounded setting, if the rocessing time j and the delivery time q j of every job J j satisfy some restricted roerties, Yuan et al. [48], Tian et al. [43], and Fang ffiffi et al. [10] roosed best ossible online algorithms with cometitive ratio þ1. Later, for general case, Tian et al. [44] imroved the cometitive ratio to ffiffi 1. It still leaves a ga between the lower bound and the uer bound. Fang et al. [10] studied the roblem on m arallel-batch machines. They used the lower bound 1 þ a m where a m þ ma m ¼ 1, which was resented by Liu et al. [7] and Tian et al. [40]. Fang et al. [11] designed an online algorithm with cometitive ratio 1: þ oð1þ (no greater than ), and then Liu et al. [8] resented a new algorithm imroving the Table For minimizing D max Problem Lower bound Uer bound References 1jonline; -batch; b ¼1; q j jd max ffiffi þ1 [39] 1jonline; -batch; b\n; q j jd max ffiffi þ1 3 [39] 1jonline; -batch; b; j ¼ ; q j jd max ffiffi ffiffi þ1 [39] ffiffi 1jonline; -batch; b ¼1; q j jd max þ1 ffiffiffi 1 [44] ffiffi ffiffi 1jonline; -batch; b ¼1; Agreeableð j ; q j ÞjD max þ1 [48] 1jonline; -batch; b ¼1; j > q j jd max ffiffi ffiffi þ1 [48] 1jonline; -batch; b ¼1; j 6 q j jd max ffiffi ffiffi þ1 [43] 1jonline; ffiffi ffiffi ffiffi -batch; b ¼1; j ½; þ1 ŠjD þ1 max [10] Pmjonline; -batch; b ¼1; j jd max 1 þ a m 1: þ oð1þ [11] Pmjonline; -batch; b ¼1; j jd max 1 þ a m 1 þ ffiffiffi b m c [8] Pmjonline; -batch; b ¼1; Agreeableð j ; q j ÞjD max 1 þ a m 1 þ a m [8] 13

8 4 J. Tian et al. uer bound to 1 þ ffiffiffi b m c. The latter authors also roosed a best ossible online algorithm for a restricted version. The main results of online scheduling on arallel-batch machines with delivery time are summarized in Table. In this art, we resent two oen roblems for future research. (1) 1jonline; -batch; b ¼1; q j jd max. () Pmjonline; -batch; b ¼1; q j jd max. 4 Other Results Li and Yuan [4] studied the online scheduling on m unbounded arallel-batch machines to minimize maximum flow time. They claimed that no online algorithm has a cometitive ratio less than 1 þ k m, where k m þðmþ1þk m ¼ 1: An online algorithm with a cometitive ratio of 1 þ 1 m was roosed. When all jobs have equal rocessing time, they designed a best ossible algorithm with a cometitive ratio of 1 þ k m. Li et al. [3] investigated incomatible families on multile batch machines where each batch has a caacity b with b ¼1 under unbounded setting and b\1 under bounded setting. The objective function is to maximize the weighted number of early jobs. They assumed that the rocessing time of each job J is equal-length integer, and the release time and the deadline are integer too. When the rocessing time is of unit length, they first established a lower bound 1 b and then rovided a greedy online algorithm with a cometitive ratio of. Then, the algorithm is best ossible when b ¼1. If the rocessing time is an arbitrary integer, they resented a ð3 þ ffiffi Þ-cometitive algorithm for both b ¼1 and b\1, which is the first algorithm with constant cometitive ratio for this roblem. To minimize the total weighted comletion times of jobs on a single batch machine under online setting, for the unbounded case, Chen et al. [7] rovided a linear-time online algorithm with a cometitive ratio of 10 3 and a randomized online algorithm with a cometitive ratio of.89. For the bounded model, they gave an algorithm with a cometitive ratio of 4 þ and a randomized online algorithm with cometitive ratio of :89 þ for any [ 0. Acknowledgments The authors would like to thank the anonymous referee for his/her constructive comments and kind suggestions. References [1] Anderson, E.J., Potts, C.N.: Online scheduling of a single machine to minimize total weighted comletion time. Math. Oer. Res. 9, (004) 13

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