Single Parallel-batching Machine Research Articles

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.

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.

Two-stage hybrid flow shop scheduling on parallel batching machines considering a job-dependent deteriorating effect and non-identical job sizes

In this paper, we investigate a specialized two-stage hybrid flow shop scheduling problem with parallel batching machines considering a job-dependent deteriorating effect and non-identical job sizes simultaneously. A novel concept of three-dimensional wasted volume based on the job normal processing time, job size, and job deteriorating rate is first proposed. Some structural properties, as well as a heuristic algorithm, are developed to solve the single parallel batching machine scheduling problem. Since the two-stage hybrid flow shop scheduling problem is NP-hard, a hybrid EDA-DE algorithm combining estimation of distribution algorithm (EDA) and differential evolution (DE) algorithm is proposed to tackle the studied problem. In addition, the Taguchi method of design of experiments (DOE) is implemented to tune the parameters of the EDA-DE. Finally, a series of computational experiments are carried out to compare the performance of the proposed hybrid EDA-DE algorithm and some recent existing algorithms from the literature, and the comparative results validate the effectiveness and efficiency of the proposed algorithm.

Two-agent scheduling with deteriorating jobs on a single parallel-batching machine: refining computational complexity

Tang et al. (Eur J Oper Res 263:401–411, 2017) have recently introduced a parallel-batching machine scheduling problem with linearly deteriorating jobs of two agents and presented a computational complexity classification of various special cases of this problem, including a number of NP-hardness proofs. We refine these results by demonstrating strong NP-hardness of several special cases, which are proved NP-hard in the ordinary sense in Tang et al. (Eur J Oper Res 263:401–411, 2017). Our reduction employs the problem studied in the first issue of Journal of Scheduling.

Parallel-batching machines scheduling problem with a truncated time-dependent learning effect via a hybrid CS-JADE algorithm

ABSTRACTThis research investigates the parallel-batching scheduling problems with a time-dependent learning effect where the job processing time is a decreasing function of its starting time. Both the single-machine and parallel-machine circumstances are considered, and the objective is to minimize the makespan. For the single parallel-batching machine scheduling problem, some structural properties and a heuristic algorithm are developed to solve it. Since the parallel-batching machines scheduling problem is NP-hard, a hybrid CS-JADE algorithm combining improved cuckoo search algorithm (CS) and self-adaptive differential evolution (DE) is proposed to solve this problem. The computational experiments indicate that the proposed hybrid algorithm performs well both in effectiveness and efficiency.

Two-agent scheduling on a single parallel-batching machine to minimize the weighted sum of the agents’ makespans

We schedule the jobs from two agents with equal processing times and non-identical job sizes on a single parallel-batching machine. The objective is to minimize the weighted sum of the two makespans of the jobs from two agents. We give a lower bound on the optimal solution, and present a polynomial-time algorithm H for solving the problem. Furthermore, we prove that both the absolute worst-case ratio and the asymptotic worst-case ratio of algorithm H are the functions of the weight of one agent, $$\alpha$$ . Specifically, (1) for the absolute worst-case ratio, $$H(I)\le f_1(\alpha )\times OPT(I)$$ , where $$f_1(\alpha )=(3-3\alpha +3\alpha ^2)/(2-4\alpha +4\alpha ^2)\le 9/4$$ ; (2) for the asymptotic worst-case ratio, $$H(I)\le f_2(\alpha ) \times OPT(I)+4/3$$ , where $$f_2(\alpha )=(11-11\alpha +11\alpha ^2)/(9-18\alpha +18\alpha ^2)$$ and $$11/9 \le f_2(\alpha )\le 11/6$$ . The effectiveness of algorithm H is demonstrated by using a set of computational experiments. The results show that algorithm H performs well in practice and tends to perform better when facing the large-scale instances.

Competitive two-agent scheduling with deteriorating jobs on a single parallel-batching machine

We consider a scheduling problem in which the jobs are generated by two agents and have time-dependent proportional-linear deteriorating processing times. The two agents compete for a common single batching machine to process their jobs, and each agent has its own criterion to optimize. The jobs may have identical or different release dates. The batching machine can process several jobs simultaneously as a batch and the processing time of a batch is equal to the longest of the job processing times in the batch. The problem is to determine a schedule for processing the jobs such that the objective of one agent is minimized, while the objective of the other agent is maintained under a fixed value. For the unbounded model, we consider various combinations of regular objectives on the basis of the compatibility of the two agents. For the bounded model, we consider two different objectives for incompatible and compatible agents: minimizing the makespan of one agent subject to an upper bound on the makespan of the other agent and minimizing the number of tardy jobs of one agent subject to an upper bound on the number of tardy jobs of the other agent. We analyze the computational complexity of various problems by either demonstrating that the problem is intractable or providing an efficient exact algorithm for the problem. Moreover, for certain problems that are shown to be intractable, we provide efficient algorithms for certain special cases.

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.

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.

Single-machine parallel-batching scheduling with family jobs to minimize weighted number of tardy jobs

We consider the problem of scheduling n jobs in batches on a single parallel-batching machine, where the jobs are partitioned into jobs families and the jobs in each family have the same due date. The objective is to minimize the weighted number of tardy jobs. We first devise an efficient pseudo-polynomial time and a fully polynomial time approximation scheme for the weighted problem. Then we present O(n2)-time and O(nlogn)-time algorithms for the case where the jobs have the same weight and for the case where the jobs have the same processing time, respectively.

An optimal online algorithm for single parallel-batch machine scheduling with incompatible job families to minimize makespan

We consider the online scheduling of incompatible job families on an unbounded parallel-batch machine to minimize the makespan, where jobs arrive over time and the number of job families, f, is known in advance. We provide an optimal online algorithm for the problem with a competitive ratio of 1+4f2+1−12f.

An improved on-line algorithm for single parallel-batch machine scheduling with delivery times

We study an on-line single parallel-batch machine scheduling problem where each job has a processing time and a delivery time. Jobs arrive over time and the batch capacity is unbounded. Jobs can be processed in a common batch, and each job is delivered independently and immediately at its completion time on the machine. The objective is to minimize the time by which all the jobs are delivered. We provide an on-line algorithm with a competitive ratio 22−1≈1.828, which improves on a 2-competitive on-line algorithm for this problem in the literature.

Bi-criteria scheduling on a single parallel-batch machine

This paper considers bi-criteria scheduling on a single parallel-batch machine to minimizing two regular scheduling criteria that are non-decreasing in the job completion times. For minimizing the total weighted completion time and the makespan simultaneously, we prove that the problem is NP-hard in the ordinary sense and possesses a fully polynomial-time approximation scheme. For minimizing the weighted total number of tardy jobs and the makespan simultaneously, we provide a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme. Furthermore, we identify that all Pareto-optimal solutions for ∑ j = 1 n w j C j , C max and ∑ j = 1 n w j U j , C max can be found in pseudo-polynomial time, respectively.

Minimizing the makespan on a single parallel batching machine

In this paper, we consider the problem of scheduling with release dates and rejection on a single parallel batching machine, where the jobs have non-identical sizes. Our objective is to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs. First, we give a polynomial time approximation scheme (PTAS) for the case where jobs can be split. Then, we propose a 2 -approximation algorithm for the special case with identical release dates. Finally, we present an approximation algorithm for the general problem with worst-case ratio 2 + ϵ , where ϵ > 0 is an arbitrarily small constant.