Makespan minimization in online scheduling with machine eligibility

In this paper we provide a survey of online scheduling in parallel machine environments with machine eligibility constraints and the makespan as objective function. We first give a brief overview of the different parallel machine environments and then survey the various types of machine eligibility constraints, including tree-hierarchical processing sets, Grade of Service processing sets, interval processing sets, and nested processing sets. We furthermore describe the relationships between the various different types of processing sets. We proceed with describing two basic online scheduling paradigms, namely online over list and online over time. For each one of the two paradigms we survey all the results that have been recorded in the literature with regard to each type of machine eligibility constraints. We obtain also several extensions in various directions. In the concluding section we describe the most important open problems in this particular area.     

具有两个不相容工件族单位工件的有界分批在线排序问题

研究具有两个不相容工件族单位工件单机有界平行分批的在线排序问题.工件按时在线到达,目标是最小化最大完工时间.在有界平行分批排序中,容量有限制机器最多可将b个工件形成一批同时加工,每个工件及每一批的加工时间为1.不相容工件族是指来自不同工件组的工件不能放在同一批加工.对该问题提供了一个竞争比为√17+3/4的最好可能的在线算法.     

Scheduling orders on either dedicated or flexible machines in parallel to minimize total weighted completion time

We are interested in the problem of scheduling orders for different product types in a facility with a number of machines in parallel. Each order asks for certain amounts of various different product types which can be produced concurrently. Each product type can be produced on a subset of the machines. Two extreme cases of machine environments are of interest. In the first case, each product type can be produced on one and only one machine which is dedicated to that product type. In the second case, all machines are identical and flexible; each product type can be produced by any one of the machines. Moreover, when a machine in this case switches over from one product type to another, no setup is required. Each order has a release date and a weight. Preemptions are not allowed. The objective is minimizing the total weighted completion time of the orders. Even when all orders are available at time 0, both types of machine environments have been shown to be NP-hard for any fixed number (≥2) of machines. This paper focuses on the design and analysis of approximation algorithms for these two machine environments. We also present empirical comparisons of the various algorithms. The conclusions from the empirical analyses provide insights into the trade-offs with regard to solution quality, speed, and memory space. Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users. This research is supported by the National Science Foundation through grants DMI-0300156 and DMI-0245603.     

A note on “A mixed integer programming model for advanced planning and scheduling (APS)”

In a recent paper, Chen and Ji [Chen, K., Ji, P., 2007. A mixed integer programming model for advanced planning and scheduling (APS). European Journal of Operational Research 181, 515–522] develop a mixed integer programming model for advanced planning and scheduling problem that considers capacity constraints and precedence relations between the operations. The orders require processing of several operations on eligible machines. The model presented in the above paper works for the case where each operation can be processed on only one machine. However, machine eligibility means that only a subset of machines are capable of processing a job and this subset may include more than one machine. We provide a general model for advanced planning and scheduling problems with machine eligibility. Our model can be used for problems where there are alternative machines that an operation can be assigned to.     

并行分批排序综述

并行分批排序起源于半导体芯片制造过程。在并行分批排序中,工件可成批加工,批加工机器最多可同时加工B个工件,批的加工时间为批中所有工件的最大工时。首先根据传统的机器环境和目标函数对并行分批排序已有成果进行分类介绍,主要为单机和平行机的机器环境,以及极小化最大完工时间、极小化总完工时间、极小化最大延迟、极小化误工工件数、极小化总延误和极小化最大延误的目标函数;然后梳理了由基本问题所衍生出来的具有新特点的16类新型并行分批排序,包括差异尺寸工件、多目标、工件加工时间或顺序存在限制、考虑费用和具有特殊机制等情况;最后展望未来的研究方向。     

Fast approximation algorithms for bi-criteria scheduling with machine assignment costs

We consider parallel machine scheduling problems where the processing of the jobs on the machines involves two types of objectives. The first type is one of two classical objective functions in scheduling theory: either the total completion time or the makespan. The second type involves an actual cost associated with the processing of a specific job on a given machine; each job-machine combination may have a different cost. Two bi-criteria scheduling problems are considered: (1) minimize the maximum machine cost subject to the total completion time being at its minimum, and (2) minimize the total machine cost subject to the makespan being at its minimum. Since both problems are strongly NP-hard, we propose fast heuristics and establish their worst-case performance bounds.     

具有前瞻区间的两个工件组单机在线排序问题

研究具有前瞻区间的两个不相容工件组单位工件单机无界平行分批在线排序问题.工件按时在线到达, 目标是最小化最大完工时间. 在无界平行分批排序中, 一台容量无限制机器可将多个工件形成一批同时加工, 每一批的加工时间等于该批中最长工件的加工时间. 具有前瞻区间是指在时刻t, 在线算法能预见到时间区间(t,t+\beta]内到达的所有工件的信息.不可相容的工件组是指属于不同组的工件不能安排在同一批中加工.对该问题提供了一个竞争比为\ 1+\alpha 的最好可能的在线算法,其中\ \alpha 是方程2\alpha^{2}+(\beta +1)\alpha +\beta -2=0的一个正根, 这里0\leq \beta <1.     

Scheduling job families on non-identical parallel machines with time constraints

This paper studies the scheduling of lots (jobs) of different product types (job family) on parallel machines, where not all machines are able to process all job families (non-identical machines). A special time constraint, associated to each job family, should be satisfied for a machine to remain qualified for processing a job family. This constraint imposes that the time between the executions of two consecutive jobs of the same family on a qualified machine must not exceed the time threshold of the family. Otherwise, the machine becomes disqualified. This problem comes from semiconductor manufacturing, when Advanced Process Control constraints are considered in scheduling problems. To solve this problem, two Mixed Integer Linear Programming models are proposed that use different types of variables. Numerical experiments show that the second model is much more effective, and that there is a trade-off between optimizing the scheduling objective and maximizing the number of machines that remain qualified for the job families. Two heuristics are also presented and studied in the numerical experiments.     

Parallel machine scheduling with machine availability and eligibility constraints

In this paper we consider the problem of scheduling n independent jobs on m identical machines incorporating machine availability and eligibility constraints while minimizing the makespan. Each machine is not continuously available at all times and each job can only be processed on specified machines. A network flow approach is used to formulate this scheduling problem into a series of maximum flow problems. We propose a polynomial time binary search algorithm to either verify the infeasibility of the problem or solve it optimally if a feasible schedule exists.     

Coordination mechanisms with hybrid local policies

We study coordination mechanisms for scheduling n jobs on m parallel machines where agents representing the jobs interact to generate a schedule. Each agent acts selfishly to minimize the completion time of his/her own job, without considering the overall system performance that is measured by a central objective. The performance deterioration due to the lack of a central coordination, the so-called price of anarchy, is determined by the maximum ratio of the central objective function value of an equilibrium schedule divided by the optimal value. In the first part of the paper, we consider a mixed local policy with some machines using the SPT rule and other machines using the LPT rule. We obtain the exact price of anarchy for the problem of minimizing the makespan and some bounds for the problem of minimizing the total completion time. In the second part of the paper, we consider parallel machine scheduling subject to eligibility constraints. We devise new local policies based on the flexibilities and the processing times of the jobs. We show that the newly devised local policies outperform both the SPT and the LPT rules.     

Mixed integer programming for scheduling flexible flow lines with limited intermediate buffers

This paper presents new mixed integer programming formulations for scheduling of a flexible flow line with blocking. The flexible flow line consists of several processing stages in series, separated by finite intermediate buffers, where each stage has one or more identical parallel machines. The line produces several different product types and each product must be processed by, at most, one machine in each stage. A product which has completed processing on a machine may remain there and block the machine until a downstream machine becomes available for processing. The objective is to determine a production schedule for all products so as to complete the products in a minimum time. The basic mixed integer programming formulations have been enhanced to model blocking scheduling with alternative processing routes where for each product a set of routes is available for processing. A reentrant flow line where a product visits a set of stages more than once is also considered. Numerical examples are presented to illustrate applications of the various models proposed.     

考虑加工速率相互干扰的平行机调度最优策略研究

随着智能互联网的应用深入、个性化消费时代的来临,制造服务企业开始注重利用网络平台为客户提供个性化的定制服务,在此过程中派生出了产品设计师可与多名客户在线同步交互的一种新型服务模式。本文根据设计师服务效率受并行服务客户数量影响的特征,将问题刻画为机器处理速度相互影响的一类平行机调度模型,以最小化总完工时间为优化目标,研究设计最优调度方案。首先,对于只有两名设计师且各自同时处理最多两个任务的情形,提出了改进的SPT调度规则,运用归纳法证明了该规则可以生成最优加工方案。其次,对改进的SPT规则进行任务分配方式的适当松驰以便更加易于操作,并证明松驰后的新分配方案保持了解的最优性。最后,将相关结论推广至多名设计师的一般情形。上述研究为个性化在线定制服务模式下的有效调度策略制定提供了良好的理论支撑。     

The wafer probing scheduling problem (WPSP)

The wafer probing scheduling problem (WPSP) is a variation of the parallel-machine scheduling problem, which has many real-world applications, particularly, in the integrated circuit (IC) manufacturing industry. In the wafer probing factories, the jobs are clustered by their product types, which must be processed on groups of identical parallel machines and be completed before the due dates. Further, the job processing time depends on the product type, and the machine setup time is sequence dependent on the orders of jobs processed. Since the wafer probing scheduling problem involves constraints on job clusters, job-cluster dependent processing time, due dates, machine capacity, and sequence dependent setup time, it is more difficult to solve than the classical parallel-machine scheduling problem. In this paper, we formulate the WPSP as an integer programming problem. We also transform the WPSP into the vehicle routing problem with time windows (VRPTW), a well-known network routing problem which has been investigated extensively. An illustrative example is given to demonstrate the proposed transformation. Based on the provided transformation, we present three efficient algorithms to solve the WPSP near-optimally.     

On the complexity of scheduling unit-time jobs with OR-precedence constraints

We present various complexity results for scheduling unit-time jobs subject to OR-precedence constraints. We prove that minimizing the total weighted completion time is strongly NP-hard, even on a single machine. In contrast, we give a polynomial-time algorithm for minimizing the makespan and the total completion time on identical parallel machines.     

Scheduling linearly shortening jobs under precedence constraints

We consider the problem of scheduling a set of dependent jobs on a single machine with the maximum completion time criterion. The processing time of each job is variable and decreases linearly with respect to the starting time of the job. Applying a uniform approach based on the calculation of ratios of expressions that describe total processing times of chains of jobs, we show basic properties of the problem. On the basis of these properties, we prove that if precedence constraints among jobs are in the form of a set of chains, a tree, a forest or a series–parallel digraph, the problem can be solved in O(n log n) time, where n denotes the number of the jobs.     

Parallel machines scheduling to minimize job tardiness and machine deteriorating cost with deteriorating jobs

This paper studies the parallel machines bi-criteria scheduling problem (PMBSP) in a deteriorating system. Sequencing and scheduling problems (SSP) have seldom considered the two phenomena concurrently. This paper discusses the parallel machines scheduling problem with the effects of machine and job deterioration. By the machine deterioration effect, we mean that each machine deteriorates at a different rate. This deterioration is considered in terms of cost which depends on the production rate, the machine’s operating characteristics and the kind of work done by each machine. Moreover, job processing times are increasing functions of their starting times and follow a simple linear deterioration. The objective functions are minimizing total tardiness and machine deteriorating cost. The problem of total tardiness on identical parallel machines is NP-hard, thus the problem with machine deteriorating cost as an additional term is also NP-hard. We propose the LP-metric method to show the importance of our proposed multi-objective problem. A metaheuristic algorithm is developed to locate optimal or near optimal solutions based on a Tabu search mechanism. Numerical examples are presented to show the efficiency of this model.     

带有机器准备时间且允许重排的在线排序

本文中, 我们考虑了带有机器准备时间且允许重排的两台平行机在线排序问题. 其目标为极小化最大完工时间. 我们研究了两种不同的模型, 并分别给出了最好可能的算法.     

Machine scheduling with resource dependent processing times

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.     

带重入的单台机排序问题

本文考虑带重入的单台机排序问题,重入是指每个工件在机器上加工不止一次.通过把重入模型转化为带平行链约束的排序问题,我们成功地获得了单机重入问题的两个目标函数的多项式时间最优算法,一个是总带权完工时间∑ωjCj,另一个是最大费用函数hmax.     

Parallel machine scheduling with additional resources: Notation,classification, models and solution methods

Majority of parallel machine scheduling studies consider machine as the only resource. However, in most real-life manufacturing environments, jobs may require additional resources, such as automated guided vehicles, machine operators, tools, pallets, dies, and industrial robots, for their handling and processing. This paper presents a review and discussion of studies on the parallel machine scheduling problems with additional resources. Papers are surveyed in five main categories: machine environment, additional resource, objective functions, complexity results and solution methods, and other important issues. The strengths and weaknesses of the literature together with open areas for future studies are also emphasized. Finally, extensions of integer programming models for two main classes of related problems are given and conclusions are drawn based on computational studies.