Parallel Batch Scheduling Research Articles

Batch Scheduling with Proportional-Linear Deterioration and Outsourcing

We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is pj=αj(A+Dt) or pj=αjt. A job is either accepted and processed in batches on a single machine by manufactures themselves or outsourced to the third party with a certain penalty having to be paid. The objective is to minimize the maximum completion time of the accepted jobs and the total penalty of the outsourced jobs. For the pj=αj(A+Dt) model, when all the jobs are released at time zero, we show that the problem is NP-hard and present a pseudo-polynomial time algorithm, respectively. For the pj=αjt model, when the jobs have distinct m (<n) release dates, we provide a dynamic programming algorithm, where n is the number of jobs.

Parallel batch scheduling with inclusive processing set restrictions and non-identical capacities to minimize makespan

We consider the problem of scheduling n jobs on m parallel batching machines with inclusive processing set restrictions and non-identical capacities. The machines differ in their functionality but have the same processing speed. The inclusive processing set restriction has the following property: the machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process. Each job is characterized by a processing time that specifies the minimum time needed to process the job, a release date before which it cannot be processed, and a machine index which is the smallest index among the machines that can process it. Each batching machine has a limited capacity and can process a batch of jobs simultaneously as long as its capacity is not violated. The capacities of the machines are non-identical. The processing time of a batch is the maximum of the processing times of the jobs belonging to it. Jobs in the same batch have a common start time and a common completion time. The goal is to find a non-preemptive schedule so as to minimize makespan (the maximum completion time). When all jobs are released at the same time, we present two fast algorithms with approximation ratios 3 and 9/4, respectively. For the general case with unequal release dates, we develop a polynomial time approximation scheme (PTAS), which is also the first PTAS even for the case with equal release dates and without processing set restrictions.

A branch and bound algorithm for scheduling unit size jobs on parallel batching machines to minimize makespan

In this paper, we present a branch and bound algorithm for the parallel batch scheduling of jobs having different processing times, release dates and unit sizes. There are identical machines with a fixed capacity and the number of jobs in a batch cannot exceed the machine capacity. All batched jobs are processed together and the processing time of a batch is given by the greatest processing time of jobs in that batch. We compare our method to a mixed integer program as well as a method from the literature that is capable of optimally solving instances with a single machine. Computational experiments show that our method is much more efficient than the other two methods in terms of solution time for finding the optimal solution.

Online parallel-batch scheduling to minimize total weighted completion time on single unbounded machine

The online parallel-batch scheduling problem on single unbounded machine to minimize total weighted job completion time is studied. For the general case of processing time, we give an online algorithm with competitive ratio 5+1. When all jobs have identical processing times, we provide an optimal online algorithm.

A parallel machine batch scheduling problem in a brewing company

This work introduces a scheduling problem which is motivated from a real situation faced by one of the largest brewing companies in Mexico. Several products must follow a brewing process which is composed by three main stages: coction, fermentation, and conditioning. In the coction stage, there are multiple unrelated parallel machines, and in the fermentation and conditioning stages, there are several heterogeneous tanks. Additionally, during the production process, some maintenance operations must be scheduled. The problem can be seen as a parallel machine batch scheduling problem with sequence-dependent setup times. Due to the complex structure of the problem, we propose a Greedy Randomized Adaptive Search Procedure to generate good quality solutions in a short computation time. Computational experiments are conducted with real and artificial instances. The solutions obtained for the real instances show that the proposed algorithm reaches better solutions than the current solutions generated by the decision maker at the brewery, and the computation time required by our algorithm is dramatically shorter than the one required by the company.

Simultaneous subtour elimination model for single-stage multiproduct parallel batch scheduling with sequence dependent changeovers

In this paper a mixed-integer linear programming (MILP) model is presented to minimize makespan of single-stage multiproduct parallel batch production with sequence dependent changeovers. The computational inefficiency and suboptimal problems are addressed by the tight and rigorous formulation of the proposed model. Subtours (subcycles) are eliminated simultaneously so that the optimal solution is obtained in one step. The proposed model is tested with two examples. The results show that the model obtains the global optimal solutions with significant improvement in solution time.

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.

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.

Particle Swarm Optimization for Chemical Multi-Stage and Parallel Batch Scheduling

This paper presents heuristic particle swarm optimization (HPSO) with continuous codes representing the batch production on the machine at each stage to enhance the ability of particle to handle constraints. Repair strategy of particle location is also presented in this paper to meet the demand for products and maximum restriction of production capacity. Based on the analysis of characters of optimal solution, all constraints are unified as time window of batch and solving method for resource-constrained project scheduling with time window is put forward. In order to verify the performance of HPSO, this paper uses random example to conduct test. The results show that HPSO is superior to optimization software and standard PSO.

Effective ACPS-Based Rescheduling of Parallel Batch Processing Machines with MapReduce

MapReduce is a highly efficient distributed and parallel computing framework, allowing users to readily manage large clusters in parallel computing. For Big data search problem in the distributed computing environment based on MapReduce architecture, in this paper we propose an Ant colony parallel search algorithm (ACPSMR) for Big data. It take advantage of the group intelligence of ant colony algorithm for global parallel search heuristic scheduling capabilities to solve problem of multi-task parallel batch scheduling with low efficiency in the MapReduce. And we extended HDFS design in MapReduce architecture, which make it to achieve effective integration with MapReduce. Then the algorithm can make the best of the scalability, high parallelism of MapReduce. The simulation experiment result shows that, the new algorithm can take advantages of cloud computing to get good efficiency when mining Big data.

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.

Parallel batch scheduling of deteriorating jobs with release dates and rejection.

We consider the unbounded parallel batch scheduling with deterioration, release dates, and rejection. Each job is either accepted and processed on a single batching machine, or rejected by paying penalties. The processing time of a job is a simple linear increasing function of its starting time. The objective is to minimize the sum of the makespan of the accepted jobs and the total penalty of the rejected jobs. First, we show that the problem is NP-hard in the ordinary sense. Then, we present two pseudopolynomial time algorithms and a fully polynomial-time approximation scheme to solve this problem. Furthermore, we provide an optimal O(nlog⁡n) time algorithm for the case where jobs have identical release dates.

Parallel-batch scheduling and transportation coordination with waiting time constraint.

This paper addresses a parallel-batch scheduling problem that incorporates transportation of raw materials or semifinished products before processing with waiting time constraint. The orders located at the different suppliers are transported by some vehicles to a manufacturing facility for further processing. One vehicle can load only one order in one shipment. Each order arriving at the facility must be processed in the limited waiting time. The orders are processed in batches on a parallel-batch machine, where a batch contains several orders and the processing time of the batch is the largest processing time of the orders in it. The goal is to find a schedule to minimize the sum of the total flow time and the production cost. We prove that the general problem is NP-hard in the strong sense. We also demonstrate that the problem with equal processing times on the machine is NP-hard. Furthermore, a dynamic programming algorithm in pseudopolynomial time is provided to prove its ordinarily NP-hardness. An optimal algorithm in polynomial time is presented to solve a special case with equal processing times and equal transportation times for each order.

Parallel-Batch Scheduling with Two Models of Deterioration to Minimize the Makespan

We consider the bounded parallel-batch scheduling with two models of deterioration, in which the processing time of the first model ispj=aj+αtand of the second model ispj=a+αjt. The objective is to minimize the makespan. We presentO(n log n)time algorithms for the single-machine problems, respectively. And we propose fully polynomial time approximation schemes to solve the identical-parallel-machine problem and uniform-parallel-machine problem, respectively.

Batch Scheduling with Deteriorating Jobs to Minimize the Total Completion Time

We consider bounded parallel-batch scheduling with proportional-linear deteriorating jobs and the objective to minimize the total completion time. We give some properties of optimal schedules for the problem and present for it a dynamic programming algorithm running in O(b 2 m 22m) time, where b is the size of a batch and m is the number of distinct deterioration rates.

A note on two-agent scheduling on an unbounded parallel-batching machine with makespan and maximum lateness objectives

This paper studies the two-agent scheduling on an unbounded parallel-batching machine. In the problem, there are two agents A and B with each having their own job sets. The jobs of a common agent can be processed in a common batch. Moreover, each agent has an objective function to be minimized. The objective function of agent A is the makespan of his jobs and the objective function of agent B is maximum lateness of his jobs. Yazdani Sabouni and Jolai [M.T. Yazdani Sabouni, F. Jolai, Optimal methods for batch processing problem with makespan and maximum lateness objectives, Appl. Math. Model. 34 (2010) 314–324] presented a polynomial-time algorithm for the problem to minimize a positive combination of the two agents’ objective functions. Unfortunately, their algorithm is incorrect. We then dwell on the problem and present a polynomial-time algorithm for finding all Pareto optimal solutions of this two-agent parallel-batching scheduling problem.

Scheduling of deteriorating jobs with release dates to minimize the maximum lateness

In this paper, we consider the problem of scheduling n deteriorating jobs with release dates on a single (batching) machine. Each job’s processing time is a simple linear function of its starting time. The objective is to minimize the maximum lateness. When the machine can process only one job at a time, we first show that the problem is NP-hard even if there are only two distinct release dates. Then we present a 2-approximation algorithm for the case where all jobs have negative due dates. Furthermore, we prove that the earliest due date (EDD) rule provides an optimal solution to the case where all jobs have agreeable release dates, due dates and deteriorating rates, and that the EDD rule gives the worst order for the general case, respectively. When the machine can process up to b(b=∞) jobs simultaneously as a batch, i.e., the unbounded parallel-batch scheduling model, we show that the problem is NP-hard and present one property of the optimal schedule for the case where all jobs have agreeable release dates and due dates.

Bounded parallel-batching scheduling with two competing agents

Bounded parallel-batching scheduling with two competing agents

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.

The unbounded parallel-batch scheduling with rejection

In this paper, we consider the unbounded parallel-batch scheduling with rejection. A job is either rejected, in which case a certain penalty has to be paid, or accepted and processed in batches on a machine. The processing time of a batch is defined as the longest processing time of the jobs contained in it. Four problems are considered: (1) to minimize the sum of the total completion time of the accepted jobs and the total rejection penalty of the rejected jobs; (2) to minimize the total completion time of the accepted jobs subject to an upper bound on the total rejection penalty of the rejected jobs; (3) to minimize the total rejection penalty of the rejected jobs subject to an upper bound on the total completion time of the accepted jobs; (4) to find the set of all the Pareto optimal schedules. We provide a polynomial-time algorithm for the first problem. Furthermore, we show that all the other three problems are binary NP-hard and present a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme for them.