Parallel Batch Machines Research Articles

Two-agent scheduling on a single parallel-batching machine with equal processing time and non-identical job sizes

We schedule the jobs from two agents on a single parallel-batching machine with equal processing time and non-identical job sizes. The objective is to minimize the makespan of the first agent subject to an upper bound on the makespan of the other agent. We show that there is no polynomial-time approximation algorithm for solving this problem with a finite worst-case ratio, unless P=NP. Then, we propose an effective algorithm LB to obtain a lower bound of the optimal solution, and two algorithms, namely, reserved-space heuristic (RSH) and dynamic-mix heuristic (DMH), to solve the two-agent scheduling problem. Finally, we evaluate the performance of the proposed algorithms with a set of computational experiments. The results show that Algorithm LB works well and tends to perform better with the increase of the number of jobs. Furthermore, our results demonstrate that RSH and DMH work well on different cases. Specifically, when the optimal makespan on the first agent exceeds the upper bound of the makespan of the other agent, RSH outperforms or equals DMH, otherwise DMH is not less favorable than RSH.

The Shortest First Coordination Mechanism for a Scheduling Game with Parallel-Batching Machines

This paper deals with a 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. The agent of a job chooses a machine for processing its own job. The aim of each agent is to minimize the completion time of its job while the system tries to minimize the maximal completion time over all jobs, the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of Nash equilibria of the mechanism and show that it has a price of anarchy 2.

Minimizing makespan for arbitrary size jobs with release times on P-batch machines with arbitrary capacities

We consider the problem of scheduling a set of arbitrary size jobs with dynamic arrival times on a set of parallel batch machines with arbitrary capacities; our goal is to minimize the makespan. We first give a mathematical model of the problem, and provide a lower bound for the objective function value. Based on different rules of batching the jobs and scheduling the batches on the machines, two meta-heuristics based on Ant Colony Optimization (ACO) are proposed to solve the problem. The performance of the proposed algorithms is evaluated and compared with existing heuristics by computational experiments. Our results show that one of the ACO algorithms consistently finds better solutions than all the others in a reasonable amount of time.

Uniform parallel batch machines scheduling considering transportation using a hybrid DPSO-GA algorithm

This paper investigates coordinated scheduling on uniform parallel batch machines with batch transportation. Jobs are characterized by different processing time and sizes, and they are first delivered to manufacturers in batches and then processed on the uniform parallel batch machines. The manufacturers are distributed in different geographic zones and there exists one parallel batch machine in each manufacturer. A mixed integer programming model is developed for the studied problem, and its objective is to minimize the makespan. In addition, the structural properties of the problem are analyzed. A hybrid algorithm combining the merits of discrete particle swarm optimization (DPSO) and genetic algorithm (GA) is proposed to solve this problem. In the hybrid algorithm, a heuristic and a local search strategy are introduced. Finally, computational experiments are conducted and the results show that the proposed hybrid algorithm can effectively and efficiently solve the problem within a reasonable time, particularly in large-scale instances.

Effective heuristics for makespan minimization in parallel batch machines with non-identical capacities and job release times

We consider the problem of scheduling a set of \begin{document} $n$ \end{document} jobs with arbitrary job sizes, processing times and release times on a set of \begin{document} $m$ \end{document} parallel batch machines with non-identical capacities; the objective is to minimize the makespan. We first present an algorithm to compute a lower bound for the optimal makespan. Based on different rules of batching the jobs and assigning the batches to the machines, several heuristics are proposed to solve the problem. The performance of the proposed heuristics is evaluated by computational experiments. The proposed heuristics are compared against the lower bound and against each other. Our results show that the one of the proposed algorithms outperforms all the other heuristics.

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 $$2-\frac{2}{3b}-\frac{1}{3\max \{m,b\}}$$2-23b-13max{m,b}.

An ACO algorithm for makespan minimization in parallel batch machines with non-identical job sizes and incompatible job families

We study the problem of scheduling a set of N jobs with non-identical job sizes from F different families on a set of M parallel batch machines; the objective is to minimize the makespan. The problem is known to be NP-hard. A meta-heuristic based on Max–Min Ant System (MMAS) is presented. The performance of the algorithm is compared with several previously studied algorithms by computational experiments. According to our results, the average distance between the solutions found by our proposed algorithm and the lower bounds is about 4% less than that of the best of all the compared algorithms, demonstrating that our algorithm outperforms the previously studied algorithms.

Hierarchical optimization with double due dates on an unbounded parallel-batching machine to minimize maximum lateness

This paper studies a hierarchical optimization problem on an unbounded parallel-batching machine, in which two objective functions are maximum lateness induced by two sets of due dates, representing different purposes of two decision-makers. By a hierarchical optimization problem, we mean the problem of optimizing the secondary criterion under the constraint that the primary criterion is optimized. A parallel-batching machine is a machine that can handle several jobs in a batch in which all jobs start and complete respectively at the same time. We present an $$O(n\log P)$$ -time algorithm and an $$O(n^3)$$ -time algorithm for this hierarchical scheduling problem, where P is the total processing time of all jobs.

Effective heuristic for makespan minimization in parallel batch machines with non-identical capacities

We consider the problem of scheduling a set of n jobs with arbitrary job sizes on a set of m parallel batch machines with non-identical capacities; the objective is to minimize the makespan. The problem is known to be NP-hard. A heuristic based on the First-Fit-Decreasing (FFD) rule is presented as well as a meta-heuristic based on Max-Min Ant System (MMAS). The performances of the two heuristics are compared with a previously studied heuristic by computational experiments. The results show that both proposed algorithms outperform the previously studied heuristic. Moreover, the MMAS heuristic obtains better solutions compared with the FFD heuristic.

Online-List Scheduling on a Single Bounded Parallel-Batch Machine to Minimize Makespan

In this paper, we consider the online-list scheduling on a single bounded parallel-batch machine to minimize makespan. In the problem, the jobs arrive online over list. The first unassigned job in the list should be assigned to a batch before the next job is released. Each batch can accommodate up to b jobs. For b = 2, we establish a lower bound 1 + γ of competitive ratio and provide an online algorithm with a competitive ratio of [Formula: see text], where γ is the positive root of γ(γ + 1)2 = 1. For b = 3, we establish a lower bound 1 + α of competitive ratio and provide an online algorithm with a competitive ratio of 2, where α is the positive root of the equation (1 + α)(1 + α2) = 2.

A note on unbounded parallel-batch scheduling

This paper revisits the scheduling problem on an unbounded parallel batch machine for minimizing a maximum cost fmax. It was reported in the literature that the decision version of the problem is solvable in O(n2+nlog⁡P) time, where n is the number of jobs and P is the total processing time of jobs. This implies that the optimization version for minimizing fmax can be solved in weakly polynomial time. But a strongly polynomial-time algorithm has not been provided for this problem. In this paper, we present an O(n4)-time algorithm for the Pareto optimization problem for minimizing Cmax and fmax, where Cmax is the maximum completion time of jobs. Consequently, the problem for minimizing fmax can also be solved in O(n4) time.

Integrated production and delivery on parallel batching machines

We consider the problem of scheduling a set of n jobs on m identical and parallel batching machines. The machines have identical capacities equal to K and the jobs have identical processing times equal to p. Job j has a size sj, a due date dj and a profit Rj. Several jobs can be batched together and processed by a machine, provided that the total size of the jobs in the batch does not exceed the machine capacity K. The company will earn a profit of Rj dollars if job j is delivered by time dj; otherwise, it earns nothing. A third party logistic (3PL) provider will be used to deliver the jobs. The 3PL provider picks up the jobs at times T1 < T2 < ⋅⋅⋅ < Tz, and vk (1 ≤ k ≤ z) vehicles will be provided for delivery at time Tk. The vehicles have identical capacities equal to C. The objective is to find a production and delivery schedule so as to maximize the total profit that the company can earn. We show that the problem is solvable in polynomial time if the jobs have identical sizes, but it becomes unary NP-hard if the jobs have different sizes. We propose heuristics for various NP-hard cases and analyze their performances.

Pareto optimization scheduling of family jobs on a p-batch machine to minimize makespan and maximum lateness

This paper studies the Pareto optimization scheduling problem of family jobs on an unbounded parallel-batching machine to minimize makespan and maximum lateness. In the problem, the jobs are partitioned into families and scheduled in batches, where each batch is a set of jobs belonging to a common family and the processing time of a batch is defined to be the longest processing time of the jobs in the batch. The objective is to find all Pareto optimal points for minimizing makespan and maximum lateness and, for each Pareto optimal point, provide a corresponding Pareto optimal schedule. We present an algorithm to solve this Pareto optimization problem. Our algorithm is of polynomial-time when the number of families is a constant.

Online Scheduling with Delivery Time on a Bounded Parallel Batch Machine with Limited Restart

We consider the online (over time) scheduling of equal length jobs on a bounded parallel batch machine with batch capacitybto minimize the time by which all jobs have been delivered with limited restart. Here, “restart” means that a running batch may be interrupted, losing all the work done on it, and jobs in the interrupted batch are then released and become independently unscheduled jobs, called restarted jobs. “Limited restart” means that a running batch which contains some restarted jobs cannot be restarted again. Whenb=2, we propose a best possible online algorithmH(b=2)with a competitive ratio of1+α, whereαis the positive solution of2α(1+α)=1. Whenb≥3, we present a best possible online algorithmH(b≥3)with a competitive ratio of1+β, whereβis the positive solution ofβ(1+β)2=1.

Scheduling and Common Due Date Assignment on a Single Parallel-Batching Machine with Batch Delivery

We consider a single parallel-batching machine scheduling problem with delivery involving both batching scheduling and common due date assignment. The orders are first processed on the single parallel-batching machine and then delivered in batches to the customers. The batching machine can process several orders at the same time. The processing time of a production batch on the machine is equal to the longest processing time of the orders assigned into this batch. A common due date for all the orders in the same delivery batch and a delivery date for each order need to be determined in order to minimize total weighted flow time. We first prove that this problem is NP hard in the strong sense. Two optimal algorithms by using dynamic programming are derived for the two special cases with a given sequence of orders on the machine and a given batching in the production part, respectively.

Scheduling a Bounded Parallel-Batching Machine with Incompatible Job Families and Rejection

We study a scheduling problem with incompatible job families and rejection on a parallel-batching machine, where the objective is to minimize the makespan of all accepted jobs plus the total penalty of all rejected jobs. We provide a polynomial-time algorithm for the case where all jobs have identical release dates and a pseudo-polynomial-time algorithm for the case where the number of distinct release dates is fixed. We also present a 2-approximation algorithm and a polynomial-time approximation scheme for the general problem.

Online Over Time Scheduling on Parallel-Batch Machines: A Survey

Online scheduling is a rapidly developed branch in scheduling theory. In this paper, we present an extensive survey for online over time scheduling on parallel-batch machines. Some open problems are proposed for further research.

Scheduling jobs with ready times and precedence constraints on parallel batch machines using metaheuristics

In this paper, we discuss a scheduling problem for parallel batch machines where the jobs have ready times. Problems of this type can be found in semiconductor wafer fabrication facilities (wafer fabs). In addition, we consider precedence constraints among the jobs. Such constraints arise, for example, in scheduling subproblems of the shifting bottleneck heuristic when complex job shop scheduling problems are tackled. We use the total weighted tardiness as the performance measure to be optimized. Hence, the problem is NP-hard and we have to rely on heuristic solution approaches. We consider a variable neighborhood search (VNS) scheme and a greedy randomized adaptive search procedure (GRASP) to compute efficient solutions. We assess the performance of the two metaheuristics based on a large set of randomly generated problem instances and based on instances from the literature. The obtained computational results demonstrate that VNS is a very fast heuristic that quickly leads to high-quality solutions, whereas the GRASP is slightly outperformed by the VNS approach. However, the GRASP approach has the advantage that it can be parallelized in a more natural manner compared to VNS.

Coordination of production and transportation in supply chain scheduling

This paper investigates a three-stage supply chain scheduling problem in the application area of aluminium production. Particularly, the first and the third stages involve two factories, i.e., the extrusion factory of the supplier and the aging factory of the manufacturer, where serial batching machine and parallel batching machine respectively process jobs in different ways. In the second stage, a single vehicle transports jobs between the two factories. In our research, both setup time and capacity constraints are explicitly considered. For the problem of minimizing the makespan, we formalize it as a mixed integer programming model and prove it to be strongly NP-hard. Considering the computational complexity, we develop two heuristic algorithms applied in two different cases of this problem. Accordingly, two lower bounds are derived, based on which the worst case performance is analyzed. Finally, different scales of random instances are generated to test the performance of the proposed algorithms. The computational results show the effectiveness of the proposed algorithms, especially for large-scale instances.

A meta-heuristic to minimize makespan for parallel batch machines with arbitrary job sizes

We consider the problem of scheduling a set of n jobs with arbitrary job sizes on a set of m identical and parallel batch machines so as to minimize the makespan. Motivated by the computational complexity of the problem, we propose a meta-heuristic based on the max–min ant system method. Computational experiments are performed with randomly generated test data. The results show that our algorithm outperforms several of the previously studied algorithms.