Single Machine Parallel Batch Scheduling Research Articles

Single machine parallel-batch scheduling under time-of-use electricity prices: New formulations and optimisation approaches

Electricity costs account for a relatively large proportion of total production costs in many energy-intensive manufacturing industries. And energy-efficient scheduling is getting increasing attention from researchers owing to its wide real-world applications and computational challenges. In this study, the single-machine parallel-batch scheduling problem with non-identical job sizes under time-of-use electricity prices is investigated. The objective is to minimise the total energy consumption cost such that the makespan does not exceed a given deadline. First, an improved time-slot-indexed formulation is proposed to eliminate symmetric solutions. Subsequently, a new set partition-based formulation is developed. To the best of our knowledge, there is no exact algorithm for this problem, except for some formulations solved by off-the-shelf solvers. To address this problem, a branch-and-price algorithm is developed with a novel and efficient branching rule. In addition, a column-generation-based heuristic is proposed to solve larger-scale instances. Extensive numerical experiments show that the efficiency of the branch-and-price algorithm is significantly better than that of off-the-shelf solvers, and many instances with up to 100 jobs could be solved to optimality within a 20-minute computational time limit for the first time. The column-generation-based heuristic can efficiently provide better solutions than existing heuristics (2.75% average gap in three minutes on instances with 100 jobs and 3.12% average gap in ten minutes on instances with 200 jobs).

On lower and upper bounds for single machine parallel batch scheduling

On lower and upper bounds for single machine parallel batch scheduling

Parallel-batch scheduling with deterioration and rejection on a single machine

The single machine parallel-batch scheduling with deteriorating jobs and rejection is considered in this paper. A job is either rejected, in which a rejection penalty should be paid, or accepted and processed on the machine. Each job’s processing time is an increasing linear function of its starting time. The machine can process any number of jobs simultaneously as a batch. The processing time of a batch is equal to the largest processing time of the jobs in the batch. The objectives are to minimize the makespan and the total weighted completion time, respectively, under the condition that the total rejection penalty cannot exceed a given upper bound Q. We show that both problems are NP-complete and present dynamic programming algorithms and fully polynomial time approximation schemes (FPTASs) for the considered problems.

Single-Machine Parallel-Batch Scheduling with Nonidentical Job Sizes and Rejection

We investigate the single-machine parallel-batch scheduling problem with nonidentical job sizes and rejection. In this problem, a set of jobs with different processing times and nonidentical sizes is given to be possibly processed on a parallel-batch processing machine. Each job is either accepted and then processed on the machine or rejected by paying its rejection penalty. Preemption is not allowed. Our task is to choose the accepted jobs and schedule them as batches on the machine to minimize the makespan of the accepted jobs plus the total rejection penalty of the rejected jobs. We provide an integer programming formulation to exactly solve our problem. Then, we propose three fast heuristic algorithms to solve the problem and evaluate their performances by using a small numerical example.

Single machine parallel-batching scheduling problem with fuzzy due-date and fuzzy precedence relation

A problem of single machine parallel-batching problem with fuzzy due-date and fuzzy precedence relation is investigated. Each job has a positive processing time. Set-up times are assumed to be identical for all batches. All batch sizes cannot exceed a common upper bound. The length of a batch is equal to the largest processing time among all jobs in the batch. Fuzzy due-date denotes the degree of satisfaction with respect to completion times of jobs. Fuzzy precedence constraint expresses the satisfaction level about precedence between two jobs. The objective is to minimise maximum completion time, maximise the minimum value of desirability of the fuzzy due-date and the minimum value of desirability of the fuzzy precedence. First, we propose a fuzzy due-date and ordinary precedence model, which maximises the minimum satisfaction degree of fuzzy due-date. An efficient iterative algorithm based on Procedure HL is designed. On that basis, another efficient algorithm to seek non-dominated solution is presented for the main problem in this paper. The non-dominated solution is defined to be consists of batch size, batch number and allocation of jobs to batches. The whole solution procedure needs at most computational time. Finally, we illustrate the procedure with a numerical example.

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.

Approximability results for the resource-constrained project scheduling problem with a single type of resources

In this paper, we consider the well-known resource-constrained project scheduling problem. We give some arguments that already a special case of this problem with a single type of resources is not approximable in polynomial time with an approximation ratio bounded by a constant. We prove that there exist instances for which the optimal makespan values for the non-preemptive and the preemptive problems have a ratio of O(logn), where n is the number of jobs. This means that there exist instances for which the lower bound of Mingozzi et al. has a bad relative error of O(logn), and the calculation of this bound is an NP-hard problem. In addition, we give a proof that there exists a type of instances for which known approximation algorithms with polynomial time complexity have an approximation ratio of at least equal to \(O(\sqrt{n})\), and known lower bounds have a relative error of at least equal to O(logn). This type of instances corresponds to the single machine parallel-batch scheduling problem 1|p−batch,b=∞|Cmax.

Bounded single-machine parallel-batch scheduling with release dates and rejection

We consider the bounded single-machine parallel-batch scheduling problem with release dates and rejection. A job is either rejected, in which case a certain penalty has to be paid, or accepted and then processed on the machine. The objective is to minimize the sum of the makespan of the accepted jobs and the total penalty of the rejected jobs. When the jobs have identical release dates, we present a polynomial-time algorithm. When the jobs have a constant number of release dates, we give a pseudo-polynomial-time algorithm. For the general problem, we provide a 2-approximation algorithm and a polynomial-time approximation scheme.

Single machine parallel-batch scheduling with deteriorating jobs

We consider several single machine parallel-batch scheduling problems in which the processing time of a job is a linear function of its starting time. We give a polynomial-time algorithm for minimizing the maximum cost, an O ( n 5 ) time algorithm for minimizing the number of tardy jobs, and an O ( n 2 ) time algorithm for minimizing the total weighted completion time. Furthermore, we prove that the problem for minimizing the weighted number of tardy jobs is binary NP-hard.

Single machine unbounded parallel-batch scheduling with forbidden intervals

In this paper we consider the single machine parallel-batch scheduling with forbidden intervals. There are some forbidden intervals in which the machine cannot be available. The jobs are processed in batches form in the remaining free time-slots without preemption, where the processing time of a batch is defined to be the maximum processing time of the jobs in this batch. We show that, when the objective is bottleneck form, maximum lateness, or makespan with release dates of jobs, the considered problem can be solved in polynomial time.

A best on-line algorithm for the single machine parallel-batch scheduling with restricted delivery times

We consider a single batch machine on-line scheduling problem with delivery times. In this paper on-line means that jobs arrive over time and the characteristics of jobs are unknown until their arrival times. Once the processing of a job is completed it is delivered to the destination. The objective is to minimize the time by which all jobs have been delivered. For each job Jj, its processing time and delivery time are denoted by pj and qj, respectively. We consider two restricted models: (1) the jobs have small delivery times, i.e., for each job Jj, qj≤pj; (2) the jobs have agreeable processing and delivery times, i.e., for any two jobs Ji and Jj, pi>pj implies qi≥qj. We provide an on-line algorithm with competitive ratio \((\sqrt{5}+1)/2\) for both problems, and the results are the best possible.

The bounded single-machine parallel-batching scheduling problem with family jobs and release dates to minimize makespan

We consider the problem of scheduling family jobs with release dates on a bounded batching machine to minimize the makespan. A polynomial-time approximation scheme for the identical job size model and an approximation algorithm with a worst-case ratio of 5 2 for the non-identical job size model will be derived.

The single-machine parallel-batching on-line scheduling problem with family jobs to minimize makespan

In this paper we consider the single-machine parallel-batching scheduling problem with family jobs under on-line setting in the sense that we construct our schedule irrevocably as time proceeds and do not know of the existence of any job until its arrival. Our objective is to minimize the maximum completion time of the jobs (makespan). We deal with two variants of the problem: the unbounded model in which the machine can handle infinite number of jobs simultaneously and the bounded model. For the unbounded case, we provide an on-line algorithm with a worst-case ratio of 2 and prove that there exists no on-line algorithm with a worst-case ratio less than 2. For the bounded case, we also present an on-line algorithm with a worst-case ratio of 2.

Single machine parallel batch scheduling problem with release dates and three hierarchical criteria to minimize makespan, machine occupation time and stocking cost

In this paper we consider an unbounded model of the single machine parallel batch scheduling problem with release dates and three hierarchical criteria. The first criterion is to minimize makespan, the second criterion is to minimize machine occupation time and the third criterion is to minimize stocking cost. We solve this problem by an O ( n 2 ) time dynamic programming algorithm.

Single machine parallel batch scheduling subject to precedence constraints

AbstractWe consider the single machine parallel batch scheduling problems to minimize makespan and total completion time, respectively, under precedence relations. The complexities of these two problems are reported as open in the literature. In this paper, we settle these open questions by showing that both problems are strongly NP‐hard, even when the precedence relations are chains. When the processing times of jobs are directly agreeable or inversely agreeable with the precedence relations, there is an O(n2) time algorithm to minimize the makespan. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O( n ( n / m +1) m ) time dynamic programming algorithm and an O( mk k +1 P 2 k −1 ) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O( n( n/ m+1) m ) time dynamic programming algorithm and an O( mk k+1 P 2 k−1 ) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.

Scheduling a batch-processing machine subject to precedence constraints, release dates and identical processing times

We consider the single machine parallel-batching scheduling problem with precedence relations, release dates and identical processing times to minimize a regular objective function. When the processing times are unit, we give an O( n 2) time optimal algorithm. When there are no precedence relations, we solve this problem by dynamic programming in O( n 3) time. When the precedence relations are “layerly complete”, we solve this problem by a dynamic programming algorithm that runs in O( n 7) time. For the total weighted completion time minimization problem, we give an O( n 2) time 3 2 -approximation algorithm. For the makespan minimization problem, we give an O( n 2) time optimal algorithm and an expression of the minimum makespan.