Heuristic algorithms for a complex parallel machine scheduling problem

In this work we present a new scheduling model for parallel machines, which extends the multiprocessor scheduling problem with release times for minimizing the total tardiness, and also extends the problem of vehicle routing with time windows. This new model is motivated by a resource allocation problem, which appears in the service sector. We present two class of heuristic algorithms for the solution of the problem, the first class is a class of greedy algorithms, the second class is based on the solutions of linear assignment problems. Furthermore we give a rescheduling algorithm, which improves a given feasible solution of the problem. This research has been supported by the Hungarian National Foundation for Scientific Research, Grant T046405.     

A comparison of branch-and-bound algorithms for a family scheduling problem with identical parallel machines

This paper investigates branch-and-bound algorithms for the problem of scheduling jobs with family setups on identical parallel machines to minimize the weighted sum of completion times. In particular, we propose a new branching scheme that appears to substantially outperform current procedures in terms of computation time and search tree size.     

List scheduling algorithms to minimize the makespan on identical parallel machines

In this paper we present constructive algorithms for the classical deterministic scheduling problem of minimizing the makespan on identical machines. Since the problem is known to beNP-hard in the strong sense, the approximate algorithms play a relevant role when solving this problem. The proposed algorithms are based on list scheduling procedures, but the assignment rule is not the same for the full set of jobs. Computational results show that these algorithms perform very well. This research has been partially supported by the Research Project H015/2000, Universidad de Alcalá. The authors are indebted to Joaquín Pérez and the referees for their helpful remarks and comments. We also wish to thank Paul Alexander Ayres for his help in the correct use of English.     

可中断的多任务平行机排序问题

Leung等(Preemptive multiprocessor order scheduling to minimize total weighted flowtime [J]. European Journal of Operational Research, 2008, 190: 40-51)研究了如下问题: 有 n 个订单, 其中每个订单 i 含有 n_i 个不同的工件. 所有的订单在零时刻已经到达, 并且工件的加工是可中断的. 每个订单 i有一个权重 \omega_i, 定义订单 i 的完工时间 C_i 为订单 i 最后一个完工工件的完工时间. 目标是找到一个可行排序使得加权总完工时间\sum\limits_{i=1}^n \omega_iC_i 最小. Leung等证明了这个问题是NP-难的, 给出了一个近似算法, 并且分析了该算法的最坏情况界. 但是定理2的证明存在一些错误. 证明了尽管定理2的证明过程存在错误, 但是其结论仍然正确. 另外, 对上述模型的一种特殊情形给出了更好的近似算法.     

Comparative evaluation of heuristic algorithms for the single machine scheduling problem with two operations per job and time-lags

This paper shows that the single machine scheduling problem with multiple operations per job separated by minimum specified time-lags is NP-hard in the strong sense. Seven simple and polynomially bounded heuristic algorithms are developed for its solution when each job requires only two operations. Empirical evaluation shows that the percentage deviation of the heuristic solutions from their lower bounds is quite low and the effectiveness of these heuristic algorithms in finding optimal schedules increases with an increase in the number of jobs.     

A note on unrelated parallel machine scheduling with time-dependent processing times

Journal of the Operational Research Society - In the literature, most of the parallel machine scheduling problems, in which the processing time of a job is a linear function of its starting time,...     

两台平行机完工时间平方和最小的排序问题

在两个竞争公司进行零和博弈过程中, 最大化两个公司收益的乘积, 在两台平行机的离线排序问题中相当于最小化两台机器完工时间的平方和. 给出了该问题修改的延缓开始\ LPT\ 算法: 首先, 将工件按照加工时间$\p_j\ $的\ LPT\ 序重新标记; 若加工时间最长的前\ $2m$\ 个工件的总加工时间\ $P(2m)< (2m+1)p_{2m+1}$, 最优的安排加工前\ $2m+1$\ 个工件, 一旦有机器空闲, 依次从第\ $2m+2$\ 个工件安排加工; 否则,\ $P(2m)\geq (2m+1)p_{2m+1}$, 最优的安排加工前\ $2m$\ 个工件, 一旦有机器空闲, 依次从第\ $2m+1$\ 个工件安排加工. 证明了该算法的最差性能比不超过\ $1+ ( \frac{1}{2m+2} )^2$, 且界是紧的.     

A note on two parallel algorithms to solve the stable marriage problem

Evidence is presented showing that the McVitie and Wilson algorithm to solve the stable marriage problem has a sequential component that is quite large on the average. Hence parallel implementations of the algorithm are likely to achieve only mediocre average case speedup. A corollary result is that an approximate solution with a few unstable pairings can be found much faster than an exact solution.     

A cyclic scheduling problem with an undetermined number of parallel identical processors

This paper presents two integer linear programming (ILP) models for cyclic scheduling of tasks with unit/general processing time. Our work is motivated by digital signal processing (DSP) applications on FPGAs (Field-Programmable Gate Arrays)—hardware architectures hosting several sets of identical arithmetic units. These hardware units can be formalized as dedicated sets of parallel identical processors. We propose a method to find an optimal periodic schedule of DSP algorithms on such architectures where the number of available arithmetic units must be determined by the scheduling algorithm with respect to the capacity of the FPGA circuit. The emphasis is put on the efficiency of the ILP models. We show the advantages of our models in comparison with common ILP models on benchmarks and randomly generated instances.     

Energetic reasoning and bin-packing problem, for bounding a parallel machine scheduling problem

This paper deals with an extension of energetic reasoning, using some efficient lower bounds of the bin-packing problem, to get tight lower bounds for the P|r _i , q _i |C _max. The link between P||C _max and bin-packing problem is well-known. Our purpose is to extend the use of efficient lower bounds of the bin-packing problem to P|r _i , q _i |C _max. We focus on some time-intervals, to compute the mandatory parts of activities within this time-interval and then to deduce an associated bin-packing instance. Thus, lower bounds of the bin-packing problem are used to get new satisfiability tests for the parallel machine problem. We also propose to extend the classical time-bound adjustments of release dates and deadlines to efficiently use bin-packing lower bounds. Experimental results that prove the efficiency of our approach on several kind of instances are reported.     

An approximate algorithm for a high-multiplicity parallel machine scheduling problem

We consider a high-multiplicity parallel machine scheduling problem where the objective is to minimize the weighted sum of completion times. We suggest an approximate algorithm and we prove that it is asymptotically exact. The algorithm exploits a convex quadratic relaxation of the problem to fix a partial schedule, consisting of most jobs, and then assigns the residual jobs following a simple and general rule. The quality of the obtained solution is evidenced by some numerical tests.     

A new constructive heuristic for the flowshop scheduling problem

This paper presents a simple constructive heuristic (HFC) for the flowshop makespan problem which is capable of producing non-permutation schedules when it deems it appropriate. HFC determines the order of any two jobs in the final schedule based on their order in all two-machine problems embedded in the problem. Computational experiments indicate that HFC performs as well as NEH which is the currently best available constructive heuristic on problems where a permutation schedule is expected to be optimal. However, HFC outperforms NEH on problems where a non-permutation schedule may be optimal.     

An approximation algorithm for identical parallel machine scheduling with resource dependent processing times

We present a (3.5+?)-approximation algorithm for a scheduling problem on identical parallel machines with the objective to mimimize the makespan. The processing times depend on the usage of a single renewable resource where at any point of time at most k units from the resource are available.     

An exact algorithm for scheduling identical coupled tasks

The coupled task problem is to schedule n jobs on one machine where each job consists of two subtasks with required delay time between them. The objective is to minimize the makespan. This problem was analyzed in depth by Orman and Potts [3]. They investigated the complexity of different cases depending on the lengths a_i and b_i of the two subtasks and the delay time L_i. -hardness proofs or polynomial algorithms were given for all cases except for the one where a_i=a, b_i=b and L_i=L. In this paper we present an exact algorithm for this problem with time complexity O(nr^2L) where holds. Therefore the algorithm is linear in the number of jobs for fixed L.Acknowledgements. The authors are grateful to Hans Kellerer who called their attention to this problem.Research was supported by DAAD exchange program 324 PPP-Ungarn.     

A class of heuristic algorithms for the routing problem

Transiated fromIssledovaniya po Prikladnoi Matematike, No. 18, 1992, pp. 38–48.     

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.     

An efficient heuristic based on machine workload for the flowshop scheduling problem with setup and removal

We are concerned in this paper with solving ann jobs,M machines flowshop scheduling problem where the objective function is the minimization of the makespan. We take into account setup, processing and removal times separately. After drawing up a synthesis of existing work which addresses this type of problems, we propose a new heuristic algorithm which is based on the machine workload to find an efficient permutation schedule. Computational experiences show that our algorithm yields excellent results, particularly when bottleneck machines are present.     

A note on minimizing the sum of squares of machine completion times on two identical parallel machines

In this short note, we address the coherence between minimizing the sum of squares of machine completion times and minimizing makespan on two identical parallel machines. We show equivalence of the two objectives and identify interesting and useful relations which allow us to transfer worst-case ratios of approximation algorithms from one problem to the other.