Improved approximation for non-preemptive single machine flow-time scheduling with an availability constraint

We study the problem of scheduling n non-preemptable jobs on a single machine which is not available for processing during a given time period. The objective is to minimize the sum of the job completion times. The best known approximation algorithm for this NP-hard problem has a relative worst-case error bound of 17.6%. We present a parametric O(nlog n)-algorithm H with which better worst-case error bounds can be obtained. The best error bound calculated for the algorithm in the paper is 7.4%. In a computational experiment, we test the algorithm with the performance guarantee set to 10.2%. It turns out that randomly generated instances with up to 1000 jobs can be solved with a mean (maximum) error of 0.31% (3.18%) and a mean (maximum) computation time of 0.8 (9.7) seconds.     

An improved heuristic for two-machine flow shop scheduling with an availability constraint and nonresumable jobs

In this paper, we deal with the two-machine flow shop scheduling problem having an unavailability interval on the first machine, and nonresumable jobs. We first present an enhancement procedure that, once applied to any arbitrary solution, produces a schedule that is at most equal 2 times the optimal makespan. We then develop an improved heuristic, with a relative worst-case error of 3/2.     

A dynamic programming algorithm for scheduling jobs in a two-machine open shop with an availability constraint

This paper studies a two-machine open shop scheduling problem with an availability constraint, ie we assume that a machine is not always available and that the processing of the interrupted job can be resumed when the machine becomes available again. We consider the makespan minimization as criterion. This problem is NP-hard. We develop a pseudo-polynomial time dynamic programming algorithm to solve the problem optimally when the machine is not available at time s>0. Then, we propose a mixed integer linear programming formulation, that allows to solve instances with up to 500 jobs optimally in less than 5?min with CPLEX solver. Finally, we show that any heuristic algorithm has a worst-case error bound of 1.     

Makespan minimization for two parallel machines with an availability constraint

In this paper, we consider a two parallel machine problem where one machine is not available during a time period. The unavailable time period is fixed and known in advance. A machine is not available probably because it needs preventive maintenance or periodical repair. The objective of the problem is to minimize the makespan. For both nonresumable and resumable cases, we partition the problem into four sub-problems, each of which is solved optimally by an algorithm. Although all the algorithms have exponential time complexities, they are quite efficient in solving large-sized problems.     

Crane scheduling with non-crossing constraint

In this paper, we examine crane scheduling for ports. This important component of port operations management is studied when the non-crossing spatial constraint, which is common to crane operations, is considered. We assume that ships can be divided into holds and that cranes can move from hold to hold but jobs are not pre-emptive, so that only one crane can work on one hold or job to complete it. Our objective is to minimize the latest completion time for all jobs. We formulate this problem as an integer programming problem. We provide the proof that this problem is NP-complete and design a branch-and-bound algorithm to obtain optimal solutions. A simulated annealing meta-heuristic with effective neighbourhood search is designed to find good solutions in larger size instances. The elaborate experimental results show that the branch-and-bound algorithm runs much faster than CPLEX and the simulated annealing approach can obtain near optimal solutions for instances of various sizes.     

On scheduling with the non-idling constraint

In this paper, we give an overview of the main results obtained on the complexity of scheduling under the non-idling constraint, i.e, when the jobs assigned to each machine must be processed with no intermediate delay. That constraint is met in practice when the cost of intermediate idle time is too high due to the idle time itself and/or the machine restarting. The non idling constraint is a strong constraint that often needs a new solving approach and most results about classical scheduling problems do not easily extend to the non-idling variant of the problem. In this survey, we mainly consider the non-idling variants of the basic scheduling problems. So, we first present basic properties, complexity results and some algorithms concerning the one-machine non-idling scheduling problem. Then we consider the $m$ -machine non idling scheduling problem. We show that a few basic problems may be solved by rather easy extensions of the algorithm solving their classical counterpart. However, the complexity status of the non idling version of quite easy polynomial basic problems remains an open question. We finally consider a more constrained version of non-idling, called the “homogeneously non idling” constraint, where for any subset of machines, the union of their busy intervals must make an interval and we present the structural property that leads to a polynomial algorithm for unit time jobs and a weak precedence. We conclude by giving some research directions that seem quite interesting to study both for theoretical and practical issues.     

Single-machine serial-batching scheduling with a machine availability constraint,position-dependent processing time,and time-dependent set-up time

This article considers the single-machine serial-batching scheduling problem with a machine availability constraint, position-dependent processing time, and time-dependent set-up time. The objective of this problem is to make the decision of batching jobs and sequencing batches to minimize the makespan. To solve the problem, three cases of machine non-availability periods are considered, and the structural properties of the optimal solution are derived for each case. Based on these structural properties, an optimization algorithm is developed and an example is proposed to illustrate this algorithm.     

Two-machine flowshop scheduling with availability constraints

The majority of the scheduling literature carries a common assumption that machines are available all the time. However, this availability assumption may not be true in real industry settings, since a machine may become unavailable during certain periods of time when, for instance, a machine breakdown or a preventive maintenance activity is scheduled. Although the problem is realistic and important, it is relatively new and unstudied. In this paper, we study the two-machine flowshop problem under the assumption that the unavailable time is known in advance. We assume that if a job cannot be finished before the next down period of a machine then the job will have to partially restart when the machine has become available again. We call our model semiresumable. Our model contains two important special cases: resumable where the job can be continued without any penalty and nonresumable where the job needs to totally restart. We study the problem where an availability constraint is imposed only on one machine as well as on both machines. We provide complexity analysis, develop a pseudo-polynomial dynamic programming algorithm to solve the problem optimally and also propose heuristic algorithms with an error bound analysis.     

Resource constraint scheduling with a fractional shared resource

We consider a scheduling problem motivated by mining in remote off-grid areas. In this model, mines have pre-assigned mineral processing jobs and their own machine for executing these jobs. Each job also needs a certain amount of electricity in order to get completed. The electricity, on the other hand, is of limited supply and must be shared between the mines. We present a mathematical formulation of the problem and a Lagrangian relaxation based heuristic. Computational results which compares our heuristic with genetic algorithm and simulated annealing are also presented.     

Order acceptance and scheduling with machine availability constraints

We consider an order acceptance and scheduling model with machine availability constraints. The manufacturer (machine) is assumed to be available to process orders only within a number of discontinuous time intervals. To capture the real-life behavior of a typical manufacturer who has restrictions of time availability to process orders, our model allows the manufacturer to reject or outsource some of the orders. When an order is rejected or outsourced, an order-dependent cost of penalty will occur. The objective is to minimize the makespan of all accepted orders plus the total penalty of all rejected/outsourced orders. We study the approximability of the model and some of its important special cases.     

Single machine scheduling with semi-resumable machine availability constraints

This paper considers the semi-resumable model of single machine scheduling with a non-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 restart after the machine has become available again. For the problem with objective of minimizing makespan, 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 approximation algorithm with worst-case ratio smaller than 2 is presented. Two special cases of the latter problem are also considered, and improved algorithms are given.     

Machine scheduling with a rate-modifying activity

Motivated by a problem commonly found in electronic assembly lines, this paper deals with the problem of scheduling jobs and a rate-modifying activity on a single machine. A rate-modifying activity is an activity that changes the production rate of the equipment under consideration. Hence the processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. The decisions under consideration are when to schedule the rate-modifying activity and the sequence of jobs to optimize some performance measure.In this paper, we develop polynomial algorithms for solving problems of minimizing makespan, and total completion time respectively. We also develop pseudo-polynomial algorithms for solving problems of total weighted completion time under the agreeable ratio assumption. We prove that the problem of minimizing maximum lateness is NP-hard and also provide a pseudo-polynomial time algorithm to solve it optimally.     

机器带不可用时间限制的简单线性恶化供应链排序问题

研究的单机供应链排序问题中, 机器有一个不可用时间限制, 工件的加工时间与恶化率及其开工时间有关, 且工件的加工不可恢复. 一个或多个完工工件可组成一个发送批由车辆发送给客户, 且在机器不可用时间限制之前完工的工件必须在限制开始之时或之前完成发送. 问题的目标是最小化总发送时间与总发送费用之和. 证明问题是NP-难的, 提出了伪多项式时间的动态规划算法. 进一步, 在确定问题目标函数值的上界及下界之后, 设计了一个完全多项式时间近似方案(FPTAS).     

Approximability of two-machine no-wait flowshop scheduling with availability constraints

We consider in this paper the two-machine no-wait flowshop scheduling problem in which each machine may have an unavailable interval. We present a polynomial time approximation scheme for the problem when the unavailable interval is imposed on only one machine, or the unavailable intervals on the two machines overlap.     

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.     

Makespan minimization for two parallel machines scheduling with a periodic availability constraint: Mathematical programming model,average-case analysis,and anomalies

A mathematical programming model is proposed for the two parallel machines scheduling problem where one machine is periodically unavailable, jobs are non-preemptive, and the objective is minimizing the makespan. The model is established by transforming the two parallel machine setting into a single machine setting. Average-case analyses of the classical Longest Processing Time first (LPT) algorithm and the List Scheduling (LS) are presented. Computational experiments show that the LPT algorithm beats the LS algorithm in all the 96 combinations of two main parameters from an average-case error point of view and that the average-case error of the LPT algorithm is less than 2% when the number of jobs is greater than twenty. Unexpectedly, there also exist instances showing that the LS algorithm may beat the LPT algorithm from the average-case error point of view.     

Scatter search for project scheduling with resource availability cost

This paper considers a project scheduling problem with the objective of minimizing resource availability costs, taking into account a deadline for the project and precedence relations among the activities. Exact methods have been proposed for solving this problem, but we are not aware of existing heuristic methods. Scatter search is used to tackle this problem, and our implementation incorporates advanced strategies such as dynamic updating of the reference set, the use of frequency-based memory within the diversification generator, and a combination method based on path relinking. We also analyze the merit of employing a subset of different types when combining solutions. Extensive computational experiments involving more than 2400 instances are performed. For small instances, the performance of the proposed procedure is compared to optimal solutions generated by an exact cutting plane algorithm and upper and lower bounds from the literature. For medium and larger size instances, we developed simple multi-start heuristics and comparative results are reported.     

Preemptive scheduling with staircase and piecewise linear resource availability

Some variations are presented for the preemptive scheduling problem on unrelated processors, one shows how nonrenewable resources with a time-varying supply may be taken into account in an extension of the two-phase method; phase 1 consists in solving an LP problem and phase 2 is the construction of the schedule; such a construction reduces to the determination of integral vectors in polyhedra defined by totally unimodular matrices. In special cases, this is simply a compatible flow problem.

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Zusammenfassung Es werden Variationen für Reihenfolgeprobleme mit Unterbrechungen betrachtet bei denen die Aufgaben mit unterschiedlicher Bearbeitungszeit auf den einzelnen Maschinen gelöst werden können. Insbesondere wird ein Problem mit nichterneuerbaren Resourcen und zeitabhängigen Nachfragen behandelt und es wird gezeigt, daß dieses Problem durch eine Erweiterung der 2-Phasenmethode gelöst werden kann. In Phase 1 wird ein LP gelöst, während in Phase 2 ein zugehöriger Schedule konstruiert wird. Diese Konstruktion erfolgt durch die Bestimmung ganzzahliger Vektoren, die Ecken eines Polyeders entsprechen, der durch eine vollständig unimodulare Matrix definiert wird. In Spezialfällen reduziert sich dies auf Flußprobleme.

     

Approximation schemes for parallel machine scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to m parallel machines with availability constraints. We consider two different models of availability constraints: the preventive model, in which the unavailability is due to preventive machine maintenance, and the fixed job model, in which the unavailability is due to a priori assignment of some of the n jobs to certain machines at certain times. Both models have applications such as turnaround scheduling or overlay computing. In both models, the objective is to minimize the total weighted completion time. We assume that m is a constant, and that the jobs are non-resumable.For the preventive model, it has been shown that there is no approximation algorithm if all machines have unavailable intervals even if w_i=p_i for all jobs. In this paper, we assume that there is one machine that is permanently available and that the processing time of each job is equal to its weight for all jobs. We develop the first polynomial-time approximation scheme (PTAS) when there is a constant number of unavailable intervals. One main feature of our algorithm is that the classification of large and small jobs is with respect to each individual interval, and thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; and (2) to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Next, we show that there is no fully polynomial-time approximation scheme (FPTAS) in this case unless P=NP.For the fixed job model, it has been shown that if job weights are arbitrary then there is no constant approximation for a single machine with 2 fixed jobs or for two machines with one fixed job on each machine, unless P=NP. In this paper, we assume that the weight of a job is the same as its processing time for all jobs. We show that the PTAS for the preventive model can be extended to solve this problem when the number of fixed jobs and the number of machines are both constants.