Parallel Batch Machines Research Articles

Polynomial time algorithms to find Pareto optimal schedules of bicriteria lot scheduling problems with splitable jobs on a single parallel-batch machine

Polynomial time algorithms to find Pareto optimal schedules of bicriteria lot scheduling problems with splitable jobs on a single parallel-batch machine

Fair scheduling on parallel batch machines with non-identical capacities

The fairness of jobs from agents is considered in an advanced manufacturing factory and the fair scheduling of jobs on parallel batch machines is studied for two competing agents. Each agent and the system aim to minimize the makespan of their own jobs and the weighted sum makespans of the two agents, respectively. A modified full-batch longest processing time (MFBLPT) algorithm is proposed to minimize the makespan, and a Kalai–Smorodinsky fairness (KSF) algorithm is proposed to obtain fair schedules. These algorithms are analysed based on worst-case ratios, and the results show that these ratios are related to the workloads and importance of agents. Computational experiments are conducted to compare the performance of algorithm KSF with that of machine-centric and agent-centric algorithms. The results show that the pursuance of fairness leads to interest losses of a system, and that a proper preference of an agent leads to good performance of algorithm KSF.

Flexible job-shop scheduling problem with parallel batch machines based on an enhanced multi-population genetic algorithm

AbstractThe flexible job-shop scheduling problem (FJSP) with parallel batch processing machine (PBM) is one of those long-standing issues that needs cutting-edge approaches. It is a recent extension of standard flexible job shop scheduling problems. Despite their wide application and prevalence in practical production, it seems that current research on these types of combinatorial optimization problems remains limited and uninvestigated. More specifically, existing research mainly concentrates on the flow shop scenarios in parallel batch machines for job shop scheduling but few literature emphasis on the flexible job shop integration in these contexts. To directly address the above mentioned problems, this paper establishes an optimization model considering parallel batch processing machines, aiming to minimize the maximum completion time in operating and production environments. The proposed solution merges variable neighborhood search with multi-population genetic algorithms, conducting a neighborhood search on the elite population to reduce the likelihood of falling into local optima. Subsequently, its applicability was evaluated in computational experiments using real production scenarios from a partnering enterprise and extended datasets. The findings from the analyses indicate that the enhanced algorithm can decrease the objective value by as much as 15% compared to other standard algorithms. Importantly, the proposed approach effectively resolves flexible job shop scheduling problems involving parallel batch processing machines. The contribution of the research is providing substantial theoretical support for enterprise production scheduling.

A Cooperated Imperialist Competitive Algorithm for Unrelated Parallel Batch Machine Scheduling Problem

A Cooperated Imperialist Competitive Algorithm for Unrelated Parallel Batch Machine Scheduling Problem

Approximation algorithms for scheduling single batch machine with incompatible deteriorating jobs

Motivated by the soaking process under separate heating mode in iron and steel enterprises, we study the parallel batch machine scheduling problem with incompatible deteriorating jobs. The objective is to minimize makespan. A soaking furnace can be seen as a parallel batch processing machine. In order to avoid the thermal stress caused by excessive temperature difference, initial temperature is needed for the ingot before processing. With the increasing of waiting time, the ingot temperature decreases and the soaking time increases. This property is called deterioration. Setup time is needed between incompatible jobs. We show that if jobs have the same sizes, an optimal solution can be found withinO(nlogn) time. If jobs have identical processing times, the problem is proved to be NP-hard in the strong sense. We propose an approximate algorithm whose absolute and asymptotic worst-case ratios are less than 2 and 11/9, respectively. When the jobs have arbitrary sizes and arbitrary processing times, the model is also NP-hard in the strong sense. An approximate algorithm with an absolute and asymptotic worst-case ratio less than 2 is proposed. The time complexity isO(nlogn).

Parallel-batch scheduling with rejection: Structural properties and approximation algorithms

In this paper, we consider a parallel-batch machine scheduling (PBMS) model that unifies several existing scheduling models in the extant literature. In our scheduling model, a given set of jobs with different release dates are scheduled onto a machine that can process multiple jobs simultaneously in a batch. However, the number of jobs in a batch is limited. Each job can be rejected subject to a job-dependent penalty cost, but the total penalty cost of the jobs rejected is required to be no greater than a given bound. The objective is to minimize the weighted sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs. We provide some important structural properties and develop fast approximation algorithms. We also present an efficient polynomial time approximation scheme (EPTAS) with near-linear-time complexity. Our results significantly improve both the time complexities and performance bounds of several existing algorithms and approximation schemes.

An improved algorithm on unbounded parallel-batching scheduling to minimize maximum cost and makespan

This paper studies the bicriteria problem of scheduling n jobs on a parallel-batching machine to minimize maximum cost and makespan simultaneously. A parallel-batching machine is a machine that can handle up to b jobs in a batch. The jobs in a batch start and complete respectively at the same time and the processing time of a batch is equal to the largest processing time of jobs in the batch. We consider the unbounded case. For the above bicriteria scheduling problem, we present an O(n3)-time algorithm, which improved the best known O(n4)-time algorithm, and the time complexity is the same as the special case in which maximum cost is maximum lateness. Meanwhile, our algorithm can also solve the single-criterion unbounded parallel-batching scheduling problem to minimize maximum cost in O(n3) time, which improved the best known O(n4)-time algorithm.

Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines

Abstract We consider the problem of scheduling jobs with equal lengths and arbitrary sizes on uniform parallel batch machines with different capacities. Each machine can only process the jobs whose sizes are not larger than its capacity. Several jobs can be processed as a batch simultaneously on a machine, as long as their total size does not exceed the machine’s capacity. The objective is to minimize makespan. Under a divisibility constraint, we obtain two efficient exact algorithms. For the general problem, we obtain an efficient 2-approximation algorithm. Previous work has shown that the problem cannot be approximated to within an approximation ratio better than 2, unless P = NP, even when all machines have identical speeds and capacities.

Minimizing total completion time on non-identical parallel batch machines with arbitrary release times using ant colony optimization

This paper considers the problem of scheduling a group of jobs with arbitrary release times, non-identical sizes, and different processing times on non-identical parallel batch processing machines to minimize the total completion time. A mixed-integer programming (MIP) model is firstly constructed in this paper to solve this problem. Then since the studied problem is strongly NP-hard, a modified elite ant system algorithm with the local search (MEASL) is also proposed to solve it, which is compared with several meta-heuristic algorithms and the commercial optimization solver (Gurobi) through extensive simulation experiments. Finally, the experimental results verify the effectiveness of the proposed algorithm.

Efficiency and inefficiency of Nash equilibrium for scheduling games on batching-machines with activation cost

We study two scheduling games on parallel-batching machines with activation cost, where each game involves n jobs being processed on m parallel-batching identical machines. Each job, as an agent, selects a machine (more precisely, a batch on a machine) for processing to minimize his disutility. For the former game, on the basis of previous results, we find the conditions for the existence of Nash equilibrium, and present the parametric upper bounds of price of anarchy (PoA) and the price of stability (PoS) for the minimax and minisum objectives, and provide some mixed strategy Nash equilibria for two special cases. For the latter game, we first analyze the conditions for the nonexistence of Nash equilibrium, and provide tight approximate Nash equilibrium. Then, the conditions for the existence of Nash equilibrium are studied, and the constant-bounds of PoA and PoS for the minimax objective are given.

Online Parallel-Batch Scheduling of Learning Effect Jobs with Incompatible Job Families for Prefabricated Components

In the production scheduling of prefabricated components, we study an online [Formula: see text] parallel-batch machines scheduling model considering learning effect jobs with [Formula: see text] incompatible job families to minimize the makespan in this paper, where the capacity of batch is unbounded. Job families indicate that a job must belong to some job family and jobs of distinct job families are incapable to be executed in the same batch. The information of each job including its basic processing time [Formula: see text] and release time [Formula: see text] is unknown in advance and is revealed at the instant of its arrival. Moreover, the actual processing time of job [Formula: see text] with learning effect is [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative parameters and [Formula: see text] denotes the starting time of prefabricated job [Formula: see text], respectively. When [Formula: see text], we propose an online algorithm with a competitive ratio of [Formula: see text]. Furthermore, the performance of the online algorithm is demonstrated by numerical experiments.

Total weighted tardiness for scheduling MapReduce jobs on parallel batch machines

<p style='text-indent:20px;'>Under support of industry 4.0, researchers have shown an increased interest in MapReduce scheduling problems to process big data. However, very few studies investigate MapReduce scheduling problems under parallel batch machine environment, which is also common in practice. Motivated by this, we study a parallel batch machine scheduling problem in which all the jobs are belonging to MapReduce type. The objective of the considered problem is of minimizing the total weighted tardiness. For solving this problem, we first establish a mixed integer linear programming model, and then a rule-based genetic algorithm is developed to solve it. Numerical experiments are also conducted to demonstrate the effectiveness of the proposed method.</p>

A Coordination Mechanism for Scheduling Game on Parallel-Batch Machines with Deterioration Jobs

In this study, we consider a parallel-batch machines scheduling game problem with deterioration jobs. The processing time of a job is a simple linear function of its starting time. Each of the parallel-batch machines can process up to B jobs simultaneously as a batch. The processing time of a batch is the processing time of the job with the longest deteriorating rate in the batch. All jobs in the same batch start and complete at the same time. Each job as an agent and its individual cost is the completion time of the job. We present a coordination mechanism for the scheduling game problem with social cost of minimizing the makespan in this paper, namely fully batch longest deteriorating rate. For this problem, we precisely quantify the inefficiency of Nash equilibrium by the logarithm price of anarchy. It is defined to be the ratio between the logarithm of social cost of the worst Nash equilibrium and the logarithm of social cost of an optimum schedule. In addition, we discuss the existence of Nash equilibrium and present an upper bound and lower bounds on the logarithm price of anarchy of the coordination mechanism. We show that the mechanism has a logarithm price of anarchy at most 2 − 1 / 3 m a x m , B − 2 / 3 B .

Bicriteria scheduling on an unbounded parallel-batch machine for minimizing makespan and maximum cost

This paper studies the bicriteria problem of scheduling n jobs on an unbounded parallel-batch machine. The goal is to minimize makespan and maximum cost simultaneously. When the jobs have arbitrary processing times and equal release dates, we obtain an O(n3)-time algorithm, improving the previously best known O(n4)-time algorithm. When the jobs have equal processing times and arbitrary release dates with a precedence relation, we also obtain an O(n3)-time algorithm, which is the first polynomial time algorithm for this problem.

Online scheduling on parallel-batch machines with periodic availability constraints and job delivery

We address an online scheduling problem on identical parallel-batch machines with periodic availability constraints and job delivery. A parallel-batch machine can process several jobs simultaneously in a batch without any preemption. The processing time of a batch is equal to the maximum processing time of the jobs in this batch, and the job processing times we consider are identical on all machines. Available and unavailable time intervals occur alternately on machines. The unavailable time intervals have the same start and end time on different machines. Jobs are released over time, which means that no information of the job is known in advance before its release date. After completion, job is delivered to customer. Three objective functions, namely, maximum flow time, maximum delivery time and total flow time are considered separately. For these objectives, we derive lower bounds on competitive ratios, provide online algorithms and analyze their competitive ratios.

Scheduling a single parallel-batching machine with non-identical job sizes and incompatible job families

We study the scheduling of jobs on a single parallel-batching machine with non-identical job sizes and incompatible job families. Jobs from the same family have the same processing time and can be loaded into a batch, as long as the batch size respects the machine capacity. The objective is to minimize the total weighted completion time; this common scheduling objective is equivalent with the minimization of in-process inventory when all jobs have the same release date. Our problem combines two classic combinatorial problems, namely bin packing and single machine scheduling. We develop three new mixed-integer linear-programming formulations, namely an assignment-based formulation, a time-indexed formulation (TIF), and a set-partitioning formulation (SPF). We also propose a column generation (CG) algorithm for the SPF, a branch-and-price (B&P) algorithm and a CG-based heuristic. A heuristic framework based on proximity search is also developed using the TIF. We examine how to reduce the size (number of variables) of the formulations in a preprocessing step. The SPF and B&P can solve instances with non-unit and unit job durations to optimality with up to 80 and 150 jobs within reasonable runtime limits, respectively. The proposed heuristics perform better than the methods available in the literature for the same problem.

Efficient algorithms for scheduling equal-length jobs with processing set restrictions on uniform parallel batch machines.

We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processing set. For the case of equal release times, we give efficient exact algorithms for various objective functions. For the case of unequal release times, we give efficient exact algorithms for minimizing makespan.

Bicriteria Scheduling on an Unbounded Parallel-Batch Machine for Minimizing Makespan and Maximum Cost

Bicriteria Scheduling on an Unbounded Parallel-Batch Machine for Minimizing Makespan and Maximum Cost

Polynomial-Size Models to Minimize Total Completion Time in a Parallel Batching Environment

We present a new integer linear formulation for the problem of minimizing the total completion time on a single parallel-batching machine. The new formulation is strong, in the sense that it delivers a sharp lower bound, and compact, i.e. polynomial in size, contrasted to recent successful models for the same problem that have exponential size and require to be handled by column generation. The new model is promising: combined with a rounding procedure, it allows to deliver good solutions with small, certified optimality gaps for instances with up to 50 jobs, and we believe it is susceptible of further improvements.

Online Scheduling on a Parallel Batch Machine with Delivery Times and Limited Restarts

Online Scheduling on a Parallel Batch Machine with Delivery Times and Limited Restarts