P k |fix| C max problem is a new scheduling problem based on the multiprocessor parallel job, and it is proved to be NP hard problem when k ≥3. This paper focuses on the case of k =3. Some new observations and new te...P k |fix| C max problem is a new scheduling problem based on the multiprocessor parallel job, and it is proved to be NP hard problem when k ≥3. This paper focuses on the case of k =3. Some new observations and new techniques for P 3 |fix| C max problem are offered. The concept of semi normal schedulings is introduced, and a very simple linear time algorithm Semi normal Algorithm for constructing semi normal schedulings is developed. With the method of the classical Graham List Scheduling, a thorough analysis of the optimal scheduling on a special instance is provided, which shows that the algorithm is an approximation algorithm of ratio of 9/8 for any instance of P 3|fix| C max problem, and improves the previous best ratio of 7/6 by M.X.Goemans.展开更多
In this paper,attention is paid to study an algorithm for the common due datetotal weighted tardiness problem of single machine scheduling. Anapproximation alsorithm is given. It performs well in the sense of worst-ca...In this paper,attention is paid to study an algorithm for the common due datetotal weighted tardiness problem of single machine scheduling. Anapproximation alsorithm is given. It performs well in the sense of worst-casebehaviour and its worst-case performance ratio is 2.展开更多
A hybrid two-stage flowshop scheduling problem was considered which involves m identical parallel machines at Stage 1 and a burn-in processor M at Stage 2, and the makespan was taken as the minimization objective. Thi...A hybrid two-stage flowshop scheduling problem was considered which involves m identical parallel machines at Stage 1 and a burn-in processor M at Stage 2, and the makespan was taken as the minimization objective. This scheduling problem is NP-hard in general. We divide it into eight subcases. Except for the following two subcases： （1） b≥ an, max{m, B} 〈 n; （2） a1 ≤ b ≤ an, m ≤ B 〈 n, for all other subcases, their NP-hardness was proved or pointed out, corresponding approximation algorithms were conducted and their worst-case performances were estimated. In all these approximation algorithms, the Multifit and PTAS algorithms were respectively used, as the jobs were scheduled in m identical parallel machines.展开更多
A parallel related uniform machine system consists of m machines with different processing speeds. The speed of any machine is independent on jobs. In this paper, we consider online scheduling for jobs with arbitrary ...A parallel related uniform machine system consists of m machines with different processing speeds. The speed of any machine is independent on jobs. In this paper, we consider online scheduling for jobs with arbitrary release times on the parallel uniform machine system. The jobs appear over list in terms of order. An order includes the processing size and releasing time of a job. For this model, an algorithm with competitive ratio of 12 is addressed in this paper.展开更多
This paper considers the semi-resumable model of single machine scheduling with anon-availability period. The machine is not available for processing during a given time interval.A job cannot be completed before the n...This paper considers the semi-resumable model of single machine scheduling with anon-availability period. The machine is not available for processing during a given time interval.A job cannot be completed before the non-availability period will have to partially restartafter the machine has become available again. For the problem with objective of minimizingmakespan, the tight worst-case ratio of algorithm LPT is given, and an FPTAS is also proposed.For the problem with objective of minimizing total weighted completion time, an approximationalgorithm with worst-case ratio smaller than 2 is presented. Two special cases of the latterproblem are also considered, and improved algorithms are given.展开更多
The number of tardy jobs of the single machine scheduling problem with a variable processing time is studied in accordance with the published instances of traffic transportation management engineering. It is proved ...The number of tardy jobs of the single machine scheduling problem with a variable processing time is studied in accordance with the published instances of traffic transportation management engineering. It is proved by 3 partition problem that if the problem is of ready time and common deadline constrained, its complexity is NP hard in the strong sense. Finally, a polynomial algorithm for solving unit processing time and common deadline problems is proposed.展开更多
We address a scheduling problem with job processing time compatibility and rejection on a parallel-batching machine.The processing time of each job is defined by an interval and any number of jobs can be assigned into...We address a scheduling problem with job processing time compatibility and rejection on a parallel-batching machine.The processing time of each job is defined by an interval and any number of jobs can be assigned into a batch provided that the processing time intervals of the jobs in the common batch are not disjoint.Three problems are considered:(1)minimize the sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs;(2)minimize the makespan of accepted jobs subject to an upper bound on the total rejection penalty of rejected jobs;(3)minimize the total rejection penalty of rejected jobs subject to an upper bound on the makespan of accepted jobs.We provide an O(n2)time algorithm for the first problem.Moreover,for the other two problems,we first show that they are NP-hard,and then present pseudo-polynomial time dynamic programming algorithms and fully polynomial time approximation schemes for them,respectively.展开更多
This paper considers the integrated production and delivery scheduling on a serial batch machine,in which split is allowed in the delivery of the jobs.The objective is to minimize the makespan,i.e.,the maximum deliver...This paper considers the integrated production and delivery scheduling on a serial batch machine,in which split is allowed in the delivery of the jobs.The objective is to minimize the makespan,i.e.,the maximum delivery completion time of the jobs.Lu et al.(Theor Comput Sci 572:50–57,2015)showed that this problem is strongly NP-hard,and presented a 32-approximation algorithm.In this paper,we present an improved 43-approximation algorithm for this problem.We also present a polynomial-time algorithm for the special case when all jobs have the identical weight.展开更多
In this paper,we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties.In this problem,we are given m dedicated machines in parallel and n customer orders.Each o...In this paper,we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties.In this problem,we are given m dedicated machines in parallel and n customer orders.Each order has a delivery time and consists of m product types and each product type should be manufactured on a dedicated machine.An order is either rejected,in which case a rejection penalty has to be paid,or accepted and manufactured on the m dedicated machines.The objective is to find a solution to minimize the sum of the maximum delivery completion time of the accepted orders and the penalty of the rejected orders which is determined by a submodular function.We design an LP rounding algorithm with approximation ratio of n+1 for this problem.展开更多
We study scheduling problems with rejection on parallel-machine.Each job consists of a processing time,a rejection cost,and a release date.The goal is to minimize the makespan of the jobs accepted when the total rejec...We study scheduling problems with rejection on parallel-machine.Each job consists of a processing time,a rejection cost,and a release date.The goal is to minimize the makespan of the jobs accepted when the total rejection cost is not larger than a given threshold.Firstly,we verify that these problems are NP-hard.Secondly,for the multiprocessor scheduling problem with rejection,we give a pseudo-polynomial algorithm and two fully polynomial approximation schemes(FPTAS for short)for fixed positive integer m,where m is the number of machines.For the scheduling problem with rejection and the job with non-identical release time on m machines,we also design a pseudo-polynomial algorithm and a fully polynomial approximation scheme when m is a fixed positive integer.We provide an approximation algorithm with the worst case performance 2 for arbitrary positive integer m.Finally,we discuss the online scheduling problem with rejection.We show that even if there are just two distinct arrive times for the jobs,there is not any online algorithm whose competitive ratio is constant for it.展开更多
The authors present a semi-definite relaxation algorithm for the scheduling problem with controllable times on a single machine. Their approach shows how to relate this problem with the maximum vertex-cover problem wi...The authors present a semi-definite relaxation algorithm for the scheduling problem with controllable times on a single machine. Their approach shows how to relate this problem with the maximum vertex-cover problem with kernel constraints (MKVC).The established relationship enables to transfer the approximate solutions of MKVCinto the approximate solutions for the scheduling problem. Then, they show how to obtain an integer approximate solution for MKVC based on the semi-definite relaxation and randomized rounding technique.展开更多
This paper considers a reclaimer scheduling problem in which one has to collect bulk material from stockpiles in the quay in such a way that the time used is minimized. When reclaimers are allowed to work on the same ...This paper considers a reclaimer scheduling problem in which one has to collect bulk material from stockpiles in the quay in such a way that the time used is minimized. When reclaimers are allowed to work on the same stockpile simultaneously, a fully polynomial time approximation scheme（FPTAS） is designed. Further,we present a 2-approximation algorithm in the case that any stockpile can be handled by only one reclaimer at a time. When the number of reclaimers is two, we give a 3/2-approximation algorithm. Numerical experiments show that the algorithms perform much better than our worst case analysis guarantees.展开更多
We study a scheduling problem with incompatible job families and rejection on a parallel-batching machine,where the objective is to minimize the makespan of all accepted jobs plus the total penalty of all rejected job...We study a scheduling problem with incompatible job families and rejection on a parallel-batching machine,where the objective is to minimize the makespan of all accepted jobs plus the total penalty of all rejected jobs.We provide a polynomial-time algorithm for the case where all jobs have identical release dates and a pseudo-polynomial-time algorithm for the case where the number of distinct release dates is fixed.We also present a 2-approximation algorithm and a polynomial-time approximation scheme for the general problem.展开更多
The m-machine no-wait flowshop scheduling problem is addressed where setup times are treated as separate from processing times. The objective is to minimize total tardiness. Different dispatching rules have been inves...The m-machine no-wait flowshop scheduling problem is addressed where setup times are treated as separate from processing times. The objective is to minimize total tardiness. Different dispatching rules have been investigated and three were found to be superior. Two heuristics, a simulated annealing (SA) and a genetic algorithm (GA), have been proposed by using the best performing dispatching rule as the initial solution for SA, and the three superior dispatching rules as part of the initial population for GA. Moreover, improved versions of SA and GA are proposed using an insertion algorithm. Extensive computational experiments reveal that the improved versions of SA and GA perform about 95% better than SA and GA. The improved version of GA outperforms the improved version of SA by about 3.5%.展开更多
This paper models the scheduling of one type of flexible transfer line as an integer programming problem.Since the integer program falls within the NP class of problems, then its complexity increases exponentially as ...This paper models the scheduling of one type of flexible transfer line as an integer programming problem.Since the integer program falls within the NP class of problems, then its complexity increases exponentially as theproblem size increases, making the problem intractable even for a medium size system. An approximate algorithm,whose complexity only increases algebraically with the problem size, is presented to obtain suboptimal solutions ofscheduling problems. Numerical experiments indicate that the suboptimal solution is near to the optimal solution inthe general case where the neighboring points are dense around the optimal solution in the feasible ration of the integer program.展开更多
This paper considers the scheduling problem with rejection on m identical parallel machines to minimize the maximum flow time. The authors show that this problem is NP-hard even when there is a single machine and all ...This paper considers the scheduling problem with rejection on m identical parallel machines to minimize the maximum flow time. The authors show that this problem is NP-hard even when there is a single machine and all jobs have two distinct release dates. Furthermore, the authors present a dynamic programming algorithm and two approximation algorithms to solve them.展开更多
In this note,we provide an almost tight lower bound for the scheduling problem to meet two min-sum objectives considered by Angel et al.in Oper.Res.Lett.35(1):69–73,2007.
文摘P k |fix| C max problem is a new scheduling problem based on the multiprocessor parallel job, and it is proved to be NP hard problem when k ≥3. This paper focuses on the case of k =3. Some new observations and new techniques for P 3 |fix| C max problem are offered. The concept of semi normal schedulings is introduced, and a very simple linear time algorithm Semi normal Algorithm for constructing semi normal schedulings is developed. With the method of the classical Graham List Scheduling, a thorough analysis of the optimal scheduling on a special instance is provided, which shows that the algorithm is an approximation algorithm of ratio of 9/8 for any instance of P 3|fix| C max problem, and improves the previous best ratio of 7/6 by M.X.Goemans.
文摘In this paper,attention is paid to study an algorithm for the common due datetotal weighted tardiness problem of single machine scheduling. Anapproximation alsorithm is given. It performs well in the sense of worst-casebehaviour and its worst-case performance ratio is 2.
基金Project supported by the Science and Technology Development Fund of Shanghai University(Grant No.A.10-0101-06-0017)
文摘A hybrid two-stage flowshop scheduling problem was considered which involves m identical parallel machines at Stage 1 and a burn-in processor M at Stage 2, and the makespan was taken as the minimization objective. This scheduling problem is NP-hard in general. We divide it into eight subcases. Except for the following two subcases： （1） b≥ an, max{m, B} 〈 n; （2） a1 ≤ b ≤ an, m ≤ B 〈 n, for all other subcases, their NP-hardness was proved or pointed out, corresponding approximation algorithms were conducted and their worst-case performances were estimated. In all these approximation algorithms, the Multifit and PTAS algorithms were respectively used, as the jobs were scheduled in m identical parallel machines.
文摘A parallel related uniform machine system consists of m machines with different processing speeds. The speed of any machine is independent on jobs. In this paper, we consider online scheduling for jobs with arbitrary release times on the parallel uniform machine system. The jobs appear over list in terms of order. An order includes the processing size and releasing time of a job. For this model, an algorithm with competitive ratio of 12 is addressed in this paper.
基金Supported by the National Natural Science Foundation of China(10971191, 60021201)Fundamental Research Funds for the Central Universities(2010QNA3040)the China Postdoctoral Science Foundation
文摘This paper considers the semi-resumable model of single machine scheduling with anon-availability period. The machine is not available for processing during a given time interval.A job cannot be completed before the non-availability period will have to partially restartafter the machine has become available again. For the problem with objective of minimizingmakespan, the tight worst-case ratio of algorithm LPT is given, and an FPTAS is also proposed.For the problem with objective of minimizing total weighted completion time, an approximationalgorithm with worst-case ratio smaller than 2 is presented. Two special cases of the latterproblem are also considered, and improved algorithms are given.
文摘The number of tardy jobs of the single machine scheduling problem with a variable processing time is studied in accordance with the published instances of traffic transportation management engineering. It is proved by 3 partition problem that if the problem is of ready time and common deadline constrained, its complexity is NP hard in the strong sense. Finally, a polynomial algorithm for solving unit processing time and common deadline problems is proposed.
基金Supported by Key Research Projects of Henan Higher Education Institutions(20A110037)Young Backbone Teachers Training Program of Zhongyuan University of Technology(2018XQG15)+4 种基金Outstanding Youth Foundation of Science and Technology Innovation of Henan Province(184100510004)Natural Science Foundation of Henan Education Department(16A630061)Science and Technology Program of Henan Province(182102110129)Innovation Training Program for College Students of Henan Province(S201910485026)Basic Research Projects of Key Scientific Research Projects Plan in Henan Higher Education Institutions(20zx003)。
文摘We address a scheduling problem with job processing time compatibility and rejection on a parallel-batching machine.The processing time of each job is defined by an interval and any number of jobs can be assigned into a batch provided that the processing time intervals of the jobs in the common batch are not disjoint.Three problems are considered:(1)minimize the sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs;(2)minimize the makespan of accepted jobs subject to an upper bound on the total rejection penalty of rejected jobs;(3)minimize the total rejection penalty of rejected jobs subject to an upper bound on the makespan of accepted jobs.We provide an O(n2)time algorithm for the first problem.Moreover,for the other two problems,we first show that they are NP-hard,and then present pseudo-polynomial time dynamic programming algorithms and fully polynomial time approximation schemes for them,respectively.
基金This research was supported by the National Natural Science Foundation of China(Nos.11271338,11771406,11571321,U1504103).
文摘This paper considers the integrated production and delivery scheduling on a serial batch machine,in which split is allowed in the delivery of the jobs.The objective is to minimize the makespan,i.e.,the maximum delivery completion time of the jobs.Lu et al.(Theor Comput Sci 572:50–57,2015)showed that this problem is strongly NP-hard,and presented a 32-approximation algorithm.In this paper,we present an improved 43-approximation algorithm for this problem.We also present a polynomial-time algorithm for the special case when all jobs have the identical weight.
基金the National Natural Science Foundation of China(No.11971146)the Natural Science Foundation of Hebei Province of China(Nos.A2019205089 and A2019205092)+1 种基金Hebei Province Foundation for Returnees(No.CL201714)the Graduate Innovation Grant Program of Hebei Normal University(No.CXZZSS2022053).
文摘In this paper,we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties.In this problem,we are given m dedicated machines in parallel and n customer orders.Each order has a delivery time and consists of m product types and each product type should be manufactured on a dedicated machine.An order is either rejected,in which case a rejection penalty has to be paid,or accepted and manufactured on the m dedicated machines.The objective is to find a solution to minimize the sum of the maximum delivery completion time of the accepted orders and the penalty of the rejected orders which is determined by a submodular function.We design an LP rounding algorithm with approximation ratio of n+1 for this problem.
基金supported by the National Natural Science Foundation of China(Nos.12271295 and 12371319)Shandong Province Natural Science Foundation(Nos.ZR2019MA061 and ZR2022MA038).
文摘We study scheduling problems with rejection on parallel-machine.Each job consists of a processing time,a rejection cost,and a release date.The goal is to minimize the makespan of the jobs accepted when the total rejection cost is not larger than a given threshold.Firstly,we verify that these problems are NP-hard.Secondly,for the multiprocessor scheduling problem with rejection,we give a pseudo-polynomial algorithm and two fully polynomial approximation schemes(FPTAS for short)for fixed positive integer m,where m is the number of machines.For the scheduling problem with rejection and the job with non-identical release time on m machines,we also design a pseudo-polynomial algorithm and a fully polynomial approximation scheme when m is a fixed positive integer.We provide an approximation algorithm with the worst case performance 2 for arbitrary positive integer m.Finally,we discuss the online scheduling problem with rejection.We show that even if there are just two distinct arrive times for the jobs,there is not any online algorithm whose competitive ratio is constant for it.
文摘The authors present a semi-definite relaxation algorithm for the scheduling problem with controllable times on a single machine. Their approach shows how to relate this problem with the maximum vertex-cover problem with kernel constraints (MKVC).The established relationship enables to transfer the approximate solutions of MKVCinto the approximate solutions for the scheduling problem. Then, they show how to obtain an integer approximate solution for MKVC based on the semi-definite relaxation and randomized rounding technique.
基金Supported by the National Natural Science Foundation of China(No.11371137 and No.71431004)
文摘This paper considers a reclaimer scheduling problem in which one has to collect bulk material from stockpiles in the quay in such a way that the time used is minimized. When reclaimers are allowed to work on the same stockpile simultaneously, a fully polynomial time approximation scheme（FPTAS） is designed. Further,we present a 2-approximation algorithm in the case that any stockpile can be handled by only one reclaimer at a time. When the number of reclaimers is two, we give a 3/2-approximation algorithm. Numerical experiments show that the algorithms perform much better than our worst case analysis guarantees.
基金supported in part by National Natural Science Foundation of China(NSFC,Nos.11326191,11401604,11401605 and 11171313)NSF of Henan Province(No.132300410392)the Education Department of Henan Province Natural Science Research Program(No.14A110027)。
文摘We study a scheduling problem with incompatible job families and rejection on a parallel-batching machine,where the objective is to minimize the makespan of all accepted jobs plus the total penalty of all rejected jobs.We provide a polynomial-time algorithm for the case where all jobs have identical release dates and a pseudo-polynomial-time algorithm for the case where the number of distinct release dates is fixed.We also present a 2-approximation algorithm and a polynomial-time approximation scheme for the general problem.
文摘The m-machine no-wait flowshop scheduling problem is addressed where setup times are treated as separate from processing times. The objective is to minimize total tardiness. Different dispatching rules have been investigated and three were found to be superior. Two heuristics, a simulated annealing (SA) and a genetic algorithm (GA), have been proposed by using the best performing dispatching rule as the initial solution for SA, and the three superior dispatching rules as part of the initial population for GA. Moreover, improved versions of SA and GA are proposed using an insertion algorithm. Extensive computational experiments reveal that the improved versions of SA and GA perform about 95% better than SA and GA. The improved version of GA outperforms the improved version of SA by about 3.5%.
文摘This paper models the scheduling of one type of flexible transfer line as an integer programming problem.Since the integer program falls within the NP class of problems, then its complexity increases exponentially as theproblem size increases, making the problem intractable even for a medium size system. An approximate algorithm,whose complexity only increases algebraically with the problem size, is presented to obtain suboptimal solutions ofscheduling problems. Numerical experiments indicate that the suboptimal solution is near to the optimal solution inthe general case where the neighboring points are dense around the optimal solution in the feasible ration of the integer program.
基金supported by the National Nature Science Foundation of China under Grant Nos.11426094,11271338 and U1504103
文摘This paper considers the scheduling problem with rejection on m identical parallel machines to minimize the maximum flow time. The authors show that this problem is NP-hard even when there is a single machine and all jobs have two distinct release dates. Furthermore, the authors present a dynamic programming algorithm and two approximation algorithms to solve them.
基金supported by National Science and Engineering Research Council of Canada(No.283106)Scientific Research Common Program of Beijing Municipal Commission of Education(No.KM201210005033).
文摘In this note,we provide an almost tight lower bound for the scheduling problem to meet two min-sum objectives considered by Angel et al.in Oper.Res.Lett.35(1):69–73,2007.
