An improved delayed-start LPT algorithm for a partition problem on two identical parallel machines

We propose an off-line delayed-start LPT algorithm that sequences the first (longest) 5 jobs optimally and the remaining jobs according to the LPT principle on two identical parallel machines. We show that this algorithm has a sharper tight worst-case ratio bound than the traditional LPT algorithm for the sum of squares of machine completion times minimization problem.     

Minimizing total tardiness in an unrelated parallel-machine scheduling problem

We consider a problem of scheduling n independent jobs on m unrelated parallel machines with the objective of minimizing total tardiness. Processing times of a job on different machines may be different on unrelated parallel-machine scheduling problems. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Results of computational experiments show that the suggested algorithm gives optimal solutions for problems with up to five machines and 20 jobs in a reasonable amount of CPU time.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

Online scheduling of parallel jobs on two machines is 2-competitive

We consider online scheduling of parallel jobs on parallel machines. For the problem with two machines and the objective of minimizing the makespan, we show that 2 is a tight lower bound on the competitive ratio. For the problem with m machines, we derive lower bounds using an ILP formulation.     

A branch and bound algorithm to minimize total tardiness of jobs in a two identical-parallel-machine scheduling problem with a machine availability constraint

This research focuses on the problem of scheduling jobs on two identical parallel machines that are not continuously available with the objective of minimizing total tardiness. After processing a given number of jobs, each machine requires a preventive maintenance task, during which the machine cannot process jobs. We present dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as an upper bound obtained from a heuristic algorithm. Performance of the algorithm is evaluated through a series of computational experiments on randomly generated instances and results are reported.     

Note on Shim and Kim’s lower bounds for scheduling on identical parallel machines to minimize total tardiness

This note introduces a new lower bound for the problem of scheduling on parallel identical machines to minimize total tardiness that is based on the concepts used in the two lower bounds developed by Shim and Kim [Shim, S.O., Kim, Y.D., 2007. Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research 177, 135–146]. The note shows that the new lower bound dominates the three lower bounds used in Shim and Kim’s branch-and-bound algorithm and can be used in place of these lower bounds to lower the enumeration required.     

Optimal on-line flow time with resource augmentation

We study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ? machines. We design an algorithm of competitive ratio , where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ?. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ?m machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.     

Scheduling on parallel identical machines to minimize total tardiness

This paper focuses on the problem of scheduling n independent jobs on m identical parallel machines for the objective of minimizing total tardiness of the jobs. We develop dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as upper bounds obtained from a heuristic algorithm. Computational experiments are performed on randomly generated test problems and results show that the algorithm solves problems with moderate sizes in a reasonable amount of computation time.     

具有周期维护最小化时间表长的两台平行机调度问题(英文)

本文讨论了具有周期维护的两台平行机调度问题,目标函数为最小化时间表长.设T为维护周期,t为每次对机器维护需要的时间,当t≤T/3时,本文证明了对于该问题由LPT算法得到的最坏误差界为2.     

Parallel machine scheduling with a common due window

In this paper, we consider a machine scheduling problem where jobs should be completed at times as close as possible to their respective due dates, and hence both earliness and tardiness should be penalized. Specifically, we consider the problem with a set of independent jobs to be processed on several identical parallel machines. All the jobs have a given common due window. If a job is completed within the due window, then there is no penalty. Otherwise, there is either a job-dependent earliness penalty or a job-dependent tardiness penalty depending on whether the job is completed before or after the due window. The objective is to find an optimal schedule with minimum total earliness–tardiness penalty. The problem is known to be NP-hard. We propose a branch and bound algorithm for finding an optimal schedule of the problem. The algorithm is based on the column generation approach in which the problem is first formulated as a set partitioning type formulation and then in each branch and bound iteration the linear relaxation of this formulation is solved by the standard column generation procedure. Our computational experiments show that this algorithm is capable of solving problems with up to 40 jobs and any number of machines within a reasonable computational time.     

A note on the two-stage assembly flow shop scheduling problem with uniform parallel machines

We study the problem of minimizing the makespan in a two-stage assembly flow shop scheduling problem with uniform parallel machines. This problem is a generalization of the assembly flow shop problem with concurrent operations in the first stage and a single assembly operation in the second stage. We propose a heuristic with an absolute performance bound which becomes asymptotically optimal as the number of jobs becomes very large. We show that our results slightly improve earlier results for the simpler assembly flow shop problem (without uniform machines) and for the two-stage hybrid flow shop problem with uniform machines.     

Parallel machine scheduling problems with a single server

In this paper, we consider the problem of scheduling jobs on parallel machines with setup times. The setup has to be performed by a single server. The objective is to minimize the schedule length (makespan), as well as the forced idle time. The makespan problem is known to be NP-hard even for the case of two identical parallel machines. This paper presents a pseudopolynomial algorithm for the case of two machines when all setup times are equal to one. We also show that the more general problem with an arbitrary number of machines is unary NP-hard and analyze some list scheduling heuristics for this problem. The problem of minimizing the forced idle time is known to be unary NP-hard for the case of two machines and arbitrary setup and processing times. We prove unary NP-hardness of this problem even for the case of constant setup times. Moreover, some polynomially solvable cases are given.     

A scheduling problem with job values given as a power function of their completion times

This paper deals with a problem of scheduling jobs on the identical parallel machines, where job values are given as a power function of the job completion times. Minimization of the total loss of job values is considered as a criterion. We establish the computational complexity of the problem – strong NP-hardness of its general version and NP-hardness of its single machine case. Moreover, we solve some special cases of the problem in polynomial time. Finally, we construct and experimentally test branch and bound algorithm (along with some elimination properties improving its efficiency) and several heuristic algorithms for the general case of the problem.     

极小化总完工时间的同时加工排序

讨论了并行工件同时加工排序问题,即n个同时到达的工件在m台批处理机上排序的问题.批处理机一次最多能加工B个工件.每批的加工时间等于该批中所含工件的加工时间的最大者.主要考虑B n的特殊情况,即每批可包含任意多个工件,目标函数是极小化总完工时间.首先对同型批处理机的情况给出了动态规划算法,算法的运行时间为O(m nm+1),并进一步将结论推广到同类批处理机的情况.     

Modelling and scheduling a batch-type production on identical machines

A new formulation and a near-optimal algorithm are presented for some variation of an NP-hard parallel scheduling problem with forest-type constraints and a C_max optimality criterion where multiple performing of each operation is required. The problem is formulated using a state space approach. In the algorithm the priority in which the assignment of operations to machines is made is based on the level of operation in the graph of precedence constraints. The worst-case performance bound of the algorithm is given, and the average performance is illustrated with computational examples.     

Scheduling parallel machines with inclusive processing set restrictions and job release times

We consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule. The machines have the same processing speed, but each job is compatible with only a subset of those machines. The machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process. We present an efficient algorithm for this problem with a worst-case performance ratio of 2. We also develop a polynomial time approximation scheme (PTAS) for the problem, as well as a fully polynomial time approximation scheme (FPTAS) for the case in which the number of machines is fixed.     

A note on makespan minimization in two-stage flexible flow shops with uniform machines

We consider the two-stage flexible flow shop makespan minimization problem with uniform parallel machines. Soewandi and Elmaghraby [Soewandi, H., Elmaghraby, S., 2003. Sequencing on two-stage hybrid flowshops with uniform machines to minimize makespan. IIE Transaction 35, 467–477] developed a heuristic (S–E) and derived a machine speed-dependent worst-case ratio bound for it. We point out that this bound works well when the uniform machines have approximately equal speeds but is not indicative of the performance of the S–E heuristic when the machine speeds are in a wide range. Motivated by this observation, we propose an alternative tight machine-speed dependent worst-case bound for the S–E heuristic that works well when the machine speeds vary significantly. We then combine the two speed-dependent ratio bounds into a speed-independent bound. Our findings facilitate the narrowing of the gap between experimental performance and worst-case bound for the S–E heuristic.     

带服务器的三台平行机排序问题的复杂性和近似算法

本文研究了带服务器的三台平行机排序问题的复杂性，并给出了一个最好的在线近似算法．     

带机器准备时间的两台同型机复合半在线排序问题

本文研究了预知两种信息,带机器准备时间的两台同型平行机复合半在线排序问题,即已知所有工件加工时间总和和工件按加工时间非增顺序到达,目标为极小化最大机器完工时间的半在线排序模型.我们分析了它的下界,并给出了竞争比为7/6的最优算法.