An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

An asymptotically optimal algorithm for large-scale mixed job shop scheduling to minimize the makespan

This paper considers the large-scale mixed job shop scheduling problem with general number of jobs on each route. The problem includes ordinary machines, batch machines (with bounded or unbounded capacity), parallel machines, and machines with breakdowns. The objective is to find a schedule to minimize the makespan. For the problem, we define a virtual problem and a corresponding virtual schedule, based on which our algorithm TVSA is proposed. The performance analysis of the algorithm shows the gap between the obtained solution and the optimal solution is O(1), which indicates the algorithm is asymptotically optimal.     

A coordination mechanism for a scheduling game with parallel-batching machines

In this paper we consider the scheduling problem with parallel-batching machines from a game theoretic perspective. There are m parallel-batching machines each of which can handle up to b jobs simultaneously as a batch. The processing time of a batch is the time required for processing the longest job in the batch, and all the jobs in a batch start and complete at the same time. There are n jobs. Each job is owned by a rational and selfish agent and its individual cost is the completion time of its job. The social cost is the largest completion time over all jobs, the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of pure Nash Equilibria and offer upper and lower bounds on the price of anarchy of the coordination mechanism. We show that the mechanism has a price of anarchy no more than \(2-\frac{2}{3b}-\frac{1}{3\max \{m,b\}}\).     

大规模集成电路预烧作业中分批排序问题的数学模型

分批排序(Batch Scheduling)是在半导体生产过程的最后阶段提炼出来的一类重要的排序问题。单机分批排序问题就是n个工件在一台机器上加工,要将工件分批,每批最多可以同时加工B个工件,每批的加工时间等于此批工件中的最大的加工时间。Skutella^[8]1998年把平行机排序的P｜｜∑ω_jC_j和R｜｜∑ω_jC_j表述成二次的0-1整数规划,得到一些令人满意的结果;国内罗守成等^[9]、张倩^[10]给出了单机排序问题1｜｜∑ω_jC_j的数学规划表示,对于用数学规划来研究排序问题是一个很有意义的进展。本文首先介绍总完工时间和最小的带权单机分批排序问题1｜B｜∑ω_jC_j,然后将1｜B｜∑ω_jC_j表示成数学规划的形式,并且用数学规划中的对偶理论证明了SPT序是其特殊情况1｜B=1｜∑C_j的最优解。     

Scheduling dedicated jobs with variative processing times

We consider the following optimization problem. There is a set of \(n\) dedicated jobs that are to be processed on \(m\) parallel machines. The job set is partitioned into subsets and jobs of each subset have a common due date. Processing times of jobs are interconnected and they are the subject of the decision making. The goal is to choose a processing time for each job in a feasible way and to construct a schedule that minimizes the maximum lateness. We show that the problem is NP-hard even if \(m=1\) and that it is NP-hard in the strong sense if \(m\) is a variable. We prove that there is no approximate polynomial algorithm with guaranteed approximation ratio less than 2. We propose an integer linear formulation for the problem and perform experiments. The experiments show that the solutions obtained with CPLEX within the limit of 5 min are on average about 5 % from the optimum value for instances with up to 150 jobs, 16 subsets and 11 machines. Most instances were solved to optimality and the average CPLEX running time was 32 s for these instances.     

考虑分时电价的多目标批调度问题蚁群算法求解

对同时优化电力成本和制造跨度的多目标批处理机调度问题进行了研究,设计了两种多目标蚁群算法,基于工件序的多目标蚁群算法(J-PACO,Job-based Pareto Ant Colony Optimization)和基于成批的多目标蚁群算法(B-PACO,Batch-based Pareto Ant Colony Optimization)对问题进行求解分析。由于分时电价中电价是时间的函数,因而在传统批调度进行批排序的基础上,需要进一步确定批加工时间点以测定电力成本。提出的两种蚁群算法分别将工件和批与时间线相结合进行调度对此类问题进行求解。通过仿真实验将两种算法对问题的求解进行了比较,仿真实验表明B-PACO算法通过结合FFLPT(First Fit Longest Processing Time)启发式算法先将工件成批再生成最终方案,提高了算法搜索效率,并且在衡量算法搜索非支配解数量的Q指标和衡量非支配集与Pareto边界接近程度的HV指标上,均优于J-PACO算法。     

Online scheduling with chain precedence constraints of equal-length jobs on parallel machines to minimize makespan

We study the online scheduling problem on m identical parallel machines to minimize makespan, i.e., the maximum completion time of the jobs, where m is given in advance and the jobs arrive online over time. We assume that the jobs, which arrive at some nonnegative real times, are of equal-length and are restricted by chain precedence constraints. Moreover, the jobs arriving at distinct times are independent, and so, only the jobs arriving at a common time are restricted by the chain precedence constraints. In the literature, a best possible online algorithm of a competitive ratio 1.3028 is given for the case \(m=2\). But the problem is unaddressed for \(m\ge 3\). In this paper, we present a best possible online algorithm for the problem with \(m\ge 3\), where the algorithm has a competitive ratio of 1.3028 for \(3\le m\le 5\) and 1.3146 for \(m\ge 6\).     

Improved algorithmic results for unsplittable stable allocation problems

The stable allocation problem is a many-to-many generalization of the well-known stable marriage problem, where we seek a bipartite assignment between, say, jobs (of varying sizes) and machines (of varying capacities) that is “stable” based on a set of underlying preference lists submitted by the jobs and machines. Building on the initial work of Dean et al. (The unsplittable stable marriage problem, 2006), we study a natural “unsplittable” variant of this problem, where each assigned job must be fully assigned to a single machine. Such unsplittable bipartite assignment problems generally tend to be NP-hard, including previously-proposed variants of the unsplittable stable allocation problem (McDermid and Manlove in J Comb Optim 19(3): 279–303, 2010). Our main result is to show that under an alternative model of stability, the unsplittable stable allocation problem becomes solvable in polynomial time; although this model is less likely to admit feasible solutions than the model proposed in McDermid and Manlove (J Comb Optim 19(3): 279–303, McDermid and Manlove 2010), we show that in the event there is no feasible solution, our approach computes a solution of minimal total congestion (overfilling of all machines collectively beyond their capacities). We also describe a technique for rounding the solution of a stable allocation problem to produce “relaxed” unsplit solutions that are only mildly infeasible, where each machine is overcongested by at most a single job.     

Optimal on-line algorithms for one batch machine with grouped processing times

In this paper, we study on-line scheduling problems on a batch machine with the assumption that all jobs have their processing times in [p, (1+φ)p], where p>0 and \(\phi=(\sqrt{5}-1)/2\). Jobs arrive over time. First, we deal with the on-line problem on a bounded batch machine with the objective to minimize makespan. A class of algorithms with competitive ratio \((\sqrt{5}+1)/2\) are given. Then we consider the scheduling on an unbounded batch machine to minimize the time by which all jobs have been delivered, and provide a class of on-line algorithms with competitive ratio \((\sqrt{5}+1)/2\). The two class of algorithms are optimal for the problems studied here.     

Parallel-machine parallel-batching scheduling with family jobs and release dates to minimize makespan

We consider the scheduling of n family jobs with release dates on m identical parallel batching machines. Each batching machine can process up to b jobs simultaneously as a batch. In the bounded model, b<n, and in the unbounded model, b=∞. Jobs from different families cannot be placed in the same batch. The objective is to minimize the maximum completion time (makespan). When the number of families is a constant, for both bounded model and unbounded model, we present polynomial-time approximation schemes (PTAS).     

Batch-Processing Scheduling with Setup Times

The problem is to minimize the total weighted completion time on a single batch-processing machine with setup times. The machine can process a batch of at most B jobs at one time, and the processing time of a batch is given by the longest processing time among the jobs in the batch. The setup time of a batch is given by the largest setup time among the jobs in the batch. This batch-processing problem reduces to the ordinary uni-processor scheduling problem when B = 1. In this paper we focus on the extreme case of B = +, i.e. a batch can contain any number of jobs. We present in this paper a polynomial-time approximation algorithm for the problem with a performance guarantee of 2. We further show that a special case of the problem can be solved in polynomial time.     

Ordinal On-Line Scheduling for Maximizing the Minimum Machine Completion Time

This paper considers the on-line problem of scheduling nonpreemptively n independent jobs on m > 1 identical and parallel machines with the objective to maximize the minimum machine completion time. It is assumed that the values of the processing times are unknown but the order of the jobs by their processing times is known in advance. We are asked to decide the assignment of all the jobs to some machines at time zero by utilizing only ordinal data rather than the actual magnitudes of jobs. Algorithms to slove the problem are called ordinal algorithms. In this paper, we give lower bounds and ordinal algorithms. We first propose an algorithm MIN which is at most -competitive for any m machine case, while the lower bound is _i=1 ^m 1/i. Both are on the order of (ln m). Furthermore, for m = 3, we present an optimal algorithm.     

Online MapReduce processing on two identical parallel machines

In this work we investigate the online over-list MapReduce processing problem on two identical parallel machines, aiming at minimizing the makespan. Jobs are revealed one by one, and each job consists of one map task and one reduce task. The map task can be arbitrarily split and processed on both machines simultaneously, while the reduce task has to be processed on a single machine and it cannot be started unless the map task has been completed. We first show that the general case of the problem reduces to the classical two machine online scheduling model with an optimal competitive ratio of 3/2. For a special case where the map task is at least as long as the reduce task, we prove that no online algorithm can be less than 4/3-competitive. An optimal Greedy algorithm with a matching competitive ratio is proposed as well.     

Online MapReduce scheduling problem of minimizing the makespan

MapReduce system is a popular big data processing framework, and the performance of it is closely related to the efficiency of the centralized scheduler. In practice, the centralized scheduler often has little information in advance, which means each job may be known only after being released. In this paper, hence, we consider the online MapReduce scheduling problem of minimizing the makespan, where jobs are released over time. Both preemptive and non-preemptive version of the problem are considered. In addition, we assume that reduce tasks cannot be parallelized because they are often complex and hard to be decomposed. For the non-preemptive version, we prove the lower bound is \(\frac{m+m(\Psi (m)-\Psi (k))}{k+m(\Psi (m)-\Psi (k))}\), higher than the basic online machine scheduling problem, where k is the root of the equation \(k=\big \lfloor {\frac{m-k}{1+\Psi (m)-\Psi (k)}+1 }\big \rfloor \) and m is the quantity of machines. Then we devise an \((2-\frac{1}{m})\)-competitive online algorithm called MF-LPT (Map First-Longest Processing Time) based on the LPT. For the preemptive version, we present a 1-competitive algorithm for two machines.     

Semi-online scheduling with combined information on two identical machines in parallel

This paper is concerned with a semi-online scheduling problem with combined information on two identical parallel machines to minimize the makespan, where all the jobs have processing times in the interval \([1,\,t]\)  \((t\ge 1)\) and the jobs arrive in non-increasing order of their processing times. The objective is to minimize the makespan. For \(t\ge 1\), we obtain a lower bound \(\max _{N=1,2,3,\ldots }\left\{ \min \{\frac{4N+3}{4N+2}\,,\frac{Nt+N+1}{2N+1}\}\right\} \) and show that the competitive ratio of the \(LS\) algorithm achieves the lower bound.     

Preemptive scheduling on two identical parallel machines with a single transporter

We consider a scheduling problem on two identical parallel machines, in which the jobs are moved between the machines by an uncapacitated transporter. In the processing preemption is allowed. The objective is to minimize the time by which all completed jobs are collected together on board the transporter. We identify the structural patterns of an optimal schedule and design an algorithm that either solves the problem to optimality or in the worst case behaves as a fully polynomial-time approximation scheme.     

考虑订单类型的两台平行批处理机在线调度模型研究

探讨了两台平行批处理机的调度决策问题,着重考虑了订单具有不同加工类型、同一批次只能加工相同类型的订单以及机器批容量有限的调度情形。针对订单实时到达且需要立即决策是否接受的实际情景,运用在线理论构建了平行机批调度在线模型。证明了该问题的竞争比下界为2Bw/（1+√Bw）,其中B和w分别表示批容量和单个订单的最大完工收益。进而设计给出了收益阈值算法PT并证明其对于订单具有紧交货期限的情形竞争比为2（1+Bw）/（1+√Bw）;对于非紧交货期限的情形,证明了修正的PT算法具有竞争比为1+2（1+Bw）/（1+√Bw）。     

Processing in a multi-machine batch manufacturing system with sequence dependent cost

Many workcells in batch manufacturing systems are populated with multiple, nonidentical machines that perform similar tasks. Because of the size of a batch when a job arrives, it may be uneconomical to set up two or more machines to process the same job simultaneously. An economic decision has to be made as regards which machine in the cell to assign the job. Likewise, many multi-operation jobs can be processed using one of several feasible operation sequences that may lead to different total manufacturing costs. The cost differences are the result of several factors, among which are processing time and cost dependencies between operations, fixturing requirements, and material handling requirements. When the workcell machine selection decision is considered along with the operation sequencing decision, determination of the best machine in a cell and the best operation sequence for the batch is a non-trivial task. In this paper, we address the problem of selecting the best machine within a cell and the best operation sequence for a batch when operation cost is machine and sequence dependent. The problem is modeled mathematically and solved using a heuristic algorithm. The performance of the algorithm is compared with that of an exact solution procedure.     

Total completion time minimization in online hierarchical scheduling of unit-size jobs

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least \(1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}\), where \(x=\sqrt{2m+4}\). We then present a greedy algorithm with tight competitive ratio of \(1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}\). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when \(m=2\), we propose an optimal online algorithm with competitive ratio of \(16\) \(/\) \(13\), which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., \(n\rightarrow \infty \).     

Semi-online scheduling on two identical machines with rejection

In this paper we consider two semi-online scheduling problems with rejection on two identical machines. A sequence of independent jobs are given and each job is characterized by its size (processing time) and its penalty, in the sense that, jobs arrive one by one and can be either rejected by paying a certain penalty or assigned to some machine. No preemption is allowed. The objective is to minimize the sum of the makespan of schedule, which is yielded by all accepted jobs and the total penalties of all rejected ones. In the first problem one can reassign several scheduled jobs in rejection tache, in the second a buffer with length k is available in rejection tache. Two optimal algorithms both with competitive ratio $\frac{3}{2}$ are presented.