Parallel Batch Processing Machines Research Articles

Minimizing makespan on parallel batch processing machines with two-dimensional rectangular jobs

This paper addresses the parallel batch processing machines scheduling problem with two-dimensional bin packing constraints. One typical application of this problem is last-mile truck delivery. A set of rectangular items (or jobs) that have delivery times are processed by a set of homogeneous vehicles (or machines). Each vehicle can deliver a batch of items at a trip following a fixed route as long as these items in the batch can be placed onto the vehicle without overlapping. The total delivery time of a batch is determined by the longest delivery time of the items in the batch. The problem is to allocate the items, which may be rotated, into batches and then assign the batches to the vehicles so as to minimize the makespan. A mixed integer programming model of the problem is formulated. A number of heuristic algorithms are proposed and compared. Some heuristics for two-dimensional packing problems are adapted to the problem. The performance of the algorithms is evaluated based on computational experiments. The experimental results show that the best-fit maximal rectangle (BFMR) heuristic is the best, in which the batch with the smallest remaining capacity is selected for an item, the empty maximal rectangle with the least space in the batch is chosen to place the item, and finally the generated batches are allocated to the vehicles according to the longest batch delivery time algorithm.

Scheduling identical parallel batch processing machines involving incompatible families with different job sizes and capacity constraints

This paper investigates the problem of scheduling jobs on parallel batch processing machines with incompatible job families, non-identical job sizes, arbitrary job release times, and machine capacity constraints. A mixed-integer linear programming (MILP) model is established to minimize makespan. A lower bound is proposed to test the solution quality obtained from heuristic and metaheuristic algorithms. Since the problem is strongly NP-hard, we propose five constructive and polynomially bounded heuristic algorithms. To improve the solutions obtained by these constructive algorithms, we describe a job family constraint-guided artificial immune system (CGAIS) based algorithm. In the proposed CGAIS algorithm, only jobs in the same family can be executed by the secondary immune response. The longest batch processing time rule is used to form the batches. The effectiveness of the proposed constructive and the CGAIS algorithms is empirically tested on the randomly generated problem instances. These computational results show that the proposed CGAIS algorithm can obtain better near-optimal solutions than the existing algorithms.

A Three-Stage ACO-Based Algorithm for Parallel Batch Loading and Scheduling Problem with Batch Deterioration and Rate-Modifying Activities

This paper addresses a batch loading and scheduling problem of minimizing the makespan on parallel batch processing machines. For batch loading, jobs with compatible families can be assigned to the same batch process even if they differ in size; however, batches can only be formed from jobs within the same family, and the batch production time is determined by the family. During the batch scheduling, the deterioration effects are continuously added to batches processed in each parallel machine so that the batch production times become deteriorated. The deteriorated processing time of batches can be recovered to the original processing times of batches by a maintenance or cleaning process of machines. In this problem, we sequentially determine the batching of jobs and the scheduling of batches. Due to the complexity of the problem, we proposed a three-stage ant colony optimization algorithm. The proposed algorithm found an optimal solution for small-sized problems and achieved near-optimal solutions and better performance than a genetic algorithm or a particle swarm optimization for large-sized problems.

Efficient algorithms to minimize makespan of the unrelated parallel batch-processing machines scheduling problem with unequal job ready times

This paper considers the minimization of makespan in the unrelated parallel batch processing machines scheduling problem with considering non-identical job size and dynamic job ready time. The considered unrelated machines have different capacity and different processing speed. Each machine processes a number of the jobs as a batch at the same time so that the machine’s capacity is not exceeded. The batch processing time and the batch ready time are equal to the largest processing time and the largest ready time of jobs in the same batch, respectively. In this paper, a Mixed Integer Linear Programming (MILP) model, two categories of the heuristic procedures (six heuristics) and a meta-heuristic algorithm are proposed to solve the problem. A lower bound is also presented by relaxing of the original problem to evaluate the quality of the proposed algorithms. The computational experiments show the performance of the proposed algorithms under the considered measures.

Machine scheduling with orientation selection and two-dimensional packing for additive manufacturing

In recent years, additive manufacturing (AM) gains increasing attention in manufacturing industries due to the growing demands, and the corresponding machine scheduling problems also attract many researchers. In this paper, we study a new unrelated parallel batch processing machine scheduling problem arose in AM, which requires to simultaneously assign parts to batches, determine the orientation of parts, pack the parts to the two-dimensional surface, and allocate the batches to machines. It is the first work to consider the orientation selection for parts in machine scheduling with the objective to minimize the makespan. To solve this problem, we first present a mixed integer linear programming model. Then, a simulated annealing algorithm with designed packing strategies based on the skyline representation of packing pattern is developed. Moreover, data structure Trie is introduced to accelerate the whole procedure and four post-optimization methods are designed to further refine solutions. Finally, a comprehensive computational study is conducted. The efficiency of our heuristic algorithms is verified and the best packing strategy for this problem is identified. The advantage of considering multiple orientations of parts in machine scheduling is demonstrated by comparisons with scenarios of fixed orientation.

A multi-objective co-evolutionary algorithm of scheduling on parallel non-identical batch machines

This paper investigates the problem scheduling a set of jobs on parallel batch processing machines with different capacities and non-identical processing powers for minimizing the makespan and the total energy consumption, where the jobs have non-identical sizes, dynamical arrival time and different processing time. To address the bi-objective optimization problem, a three-populations co-evolutionary algorithm is proposed, which is based on exploration and coordination searches among three colonies. For guaranteeing the diversity of solutions, an adaptive search strategy based on the largest angle among adjacent solutions is designed, and a new method is proposed to select ants to update pheromone trails for improving the convergence of solutions. Finally, the proposed algorithm is compared with the existing multi-objective algorithms through extensive simulated experiments, and the simulated results are statistically analyzed. And the experimental results show that the proposed algorithm outperforms all the compared algorithms, which verify the validity of the algorithm proposed in this paper.

A multi-objective evolutionary algorithm based on adaptive clustering for energy-aware batch scheduling problem

For batch scheduling problems, more and more attentions have been paid to reducing energy consumption. In this paper, a complex batch scheduling problem on parallel batch processing machines considering time-of-use electricity price is investigated to minimize makespan and total electricity cost, simultaneously. Due to NP-hardness of the studied problem, a multi-objective evolutionary algorithm based on adaptive clustering is proposed, where an improved adaptive clustering method is incorporated to mine the distribution structure of solutions, which can be used to guide the search. Moreover, a new recombination strategy based on both distribution characteristics and mating probability is designed to select individuals for mating. In addition, to better balance exploration and exploitation, the mating probability is adaptively adjusted according to historical information. The experimental results demonstrate the competitiveness of the proposed algorithm in terms of solution quality.

Bi-criteria parallel batch machine scheduling to minimize total weighted tardiness and electricity cost

A bi-criteria scheduling problem for parallel identical batch processing machines in semiconductor wafer fabrication facilities is studied. Only jobs belonging to the same family can be batched together. The performance measures are the total weighted tardiness and the electricity cost where a time-of-use (TOU) tariff is assumed. Unequal ready times of the jobs and non-identical job sizes are considered. A mixed integer linear program (MILP) is formulated. We analyze the special case where all jobs have the same size, all due dates are zero, and the jobs are available at time zero. Properties of Pareto-optimal schedules for this special case are stated. They lead to a more tractable MILP. We design three heuristics based on grouping genetic algorithms that are embedded into a non-dominated sorting genetic algorithm II framework. Three solution representations are studied that allow for choosing start times of the batches to take into account the energy consumption. We discuss a heuristic that improves a given near-to-optimal Pareto front. Computational experiments are conducted based on randomly generated problem instances. The varepsilon -constraint method is used for both MILP formulations to determine the true Pareto front. For large-sized problem instances, we apply the genetic algorithms (GAs). Some of the GAs provide high-quality solutions.

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.

Exact algorithms to minimize makespan on single and parallel batch processing machines

Batch-processing machine scheduling problem is one of the challenging problems in the machine scheduling literature where machines are capable of processing a batch of jobs simultaneously. In this paper, we tackle single and parallel batch processing machine scheduling problems with the objective of minimizing makespan. We propose a reformulation for parallel batch processing machine scheduling, which is based on decomposition in two levels, and an exact algorithm for its solution. To the best of our knowledge, there is no exact algorithm to solve this problem in the literature, except for its formulations solved by off-the-shelf solvers. In the first part of the proposed algorithm, we solve the single batch processing machine problem by a column-and-cut generation algorithm that provides a lower bound for the parallel machine problem. The second part of our proposed algorithm employs a search mechanism to find the minimum makespan for the parallel machine problem, which entails the solution of the reformulation of this problem by column generation at every iteration. The novel aspect of this column generation algorithm is the integration of batch generation and machine schedule generation in a single pricing subproblem. We test the performance of the proposed algorithms on randomly generated instances and show that, on average, they outperform the off-the-shelf solver. The major findings are that the single machine problem provides tight lower and upper bounds for the parallel machine problem, and the proposed algorithm for the parallel machine problem solves more instances to optimality than that for the single machine problem.

Ant colony optimization algorithm for total weighted completion time minimization on non-identical batch machines

This paper aims at the problem of scheduling a set of jobs with arbitrary sizes on parallel batch processing machines with arbitrary capacities. The optimization objective is to minimize the total weighted completion time of jobs. After describing the studied problem, we analyze its complexity and present a lower bound of the problem. A heuristic is provided to solve the problem firstly. Then, with the proposed first job selection strategy based on the weights of jobs, two algorithms based on the ant system and the max-min ant system, respectively, are designed to address the problem. Through extensive experiments, the performance of the proposed algorithms is compared with several state-of-the-art algorithms. The comparative results verify effectiveness and efficiency of the proposed algorithms.

Simulated annealing approach to minimise total weighted tardiness of non-identical parallel batch processing machines

Simulated annealing approach to minimise total weighted tardiness of non-identical parallel batch processing machines

Simulated annealing approach to minimise total weighted tardiness of non-identical parallel batch processing machines

A variety of industries use batch processing machines (BPMs) that can process multiple jobs simultaneously per machine. This research considers the scheduling of multiple jobs onto non-identical parallel BPMs while minimising the total weighted tardiness. The problem under study is NP-hard and solving it to optimality using a commercial solver requires long run times when the size of the problem increases. A simulated annealing (SA) approach is proposed to find good solutions for large problem instances within short run times. The results obtained are compared to two metaheuristics, particle swarm optimisation (PSO) and differential evolution (DE), proposed in the literature and a commercial solver (CPLEX). An experimental study is conducted to evaluate the different solution approaches on a set of problem instances. Based on results, the authors conclude that SA is highly effective in solving large problem instances within reasonable run times when compared to CPLEX, PSO and DE.

A strong integer programming formulation for hybrid flowshop scheduling

We consider a hybrid flowshop scheduling problem that includes parallel unrelated discrete machines or batch processing machines in different stages of a production system. The problem is motivated by a bottleneck process within the production system of a transformer producer located in the Netherlands. We develop an integer programming model that minimises the total tardiness of jobs over a finite planning horizon. Our model is applicable to a wide range of production systems organised as hybrid flowshops. We strengthen our integer program by exploiting the special properties of some constraints in our formulation. We develop a decision support system (DSS) based on our proposed optimisation model. We compare the results of our initial optimisation model with an improved formulation as well as with a heuristic that was in use at the company before the implementation of our DSS. Our results show that the improved optimisation model significantly outperforms the heuristic and the initial optimisation model in terms of both the solution time and the strength of its linear programming relaxation.

A bi-objective model for a scheduling problem of unrelated parallel batch processing machines with fuzzy parameters by two fuzzy multi-objective meta-heuristics

This paper considers a bi-objective model for a scheduling problem of unrelated parallel batch processing machines to minimize the makespan and maximum tardiness, simultaneously. Each job has a specific size and the data corresponding to its ready time, due date and processing time-dependent machine are uncertain and determined by trapezoidal fuzzy numbers. Each machine has a specific capacity, in which the number of jobs assigned to each batch on the machine does not violate the machine capacity. The batch processing time, the batch ready time and the batch due date are presented by the longest processing time, the longest ready time and the shortest due date of the jobs that belong to the batch, respectively. To determine the longest and shortest time, the method suggested by Jimenez et al.cite{c18} is used for ranking the fuzzy numbers. A bi-objective fuzzy mixed-integer linear programming model is proposed and solved by two exact methods (i.e., two-phase fuzzy and $epsilon$-constraint) for small-sized problems to obtain a set of Pareto solutions. Because the problem belongs to the class NP-hard, two meta-heuristics, namely fuzzy non-dominated sorting genetic algorithm (FNSGA-II) and fuzzy multi-objective discrete teaching–learning-based optimization (FMODTLBO), are proposed. Then, the comparison of results is illustrated to show their performances. Furthermore, a new representation of the solutions is a matrix with two rows and $N$ columns (i.e., jobs) used to assign the jobs to the batches that processed on the machines.

Integrated scheduling on parallel batch processing machines with non-identical capacities

ABSTRACTIn this article, the problem of scheduling a set of jobs on parallel batch machines with arbitrary capacities is considered. The jobs have identical processing time, non-identical sizes and unequal weights. After being processed, the jobs are delivered to the customers by the vehicles. The objective is to minimize the total weighted delivery time of the jobs. Two heuristic algorithms and an algorithm based on ant colony optimization (ACO) are presented to address the problem. Considering the relationship between the jobs and the batches, a three-level candidate list is designed for building the solutions effectively. Furthermore, based on the change of the normalized weight of the batch, the heuristic information is designed to control the search direction of the ants and improve the solution quality. The experimental results show that the performance of the proposed ACO algorithm is superior to the other algorithms compared.

Improved evolutionary algorithm for parallel batch processing machine scheduling in additive manufacturing

With the increasing prosperity of additive manufacturing, the 3D-printing shop scheduling problem has presented growing importance. The scheduling of such a shop is imperative for saving time and cost, but the problem is hard to solve, especially for simultaneous multi-part assignment and placement. This paper develops an improved evolutionary algorithm for application to additive manufacturing, by combining a genetic algorithm with a heuristic placement strategy to take into account the allocation and placement of parts integrally. The algorithm is designed also to enhance the optimisation efficiency by introducing an initialisation method based on the characteristics of the 3D printing process through the development of corresponding time calculation model. Experiments show that the developed algorithm can find better solutions compared with state-of-the-art algorithms such as simple genetic algorithm, particle swarm optimisation and heuristic algorithms.

Integrated Production and Transportation Scheduling on Parallel Batch-Processing Machines

This paper studies an integrated scheduling problem which consists of both production and transportation stages. Jobs are processed on a set of parallel batch-processing machines in the production stage, and then transported to a target customer in the transportation stage. The aim is to find an optimal integrated production and transportation schedule with the objective to minimize the weighted sum of total delivery time and total transportation cost with a relative preference α. A 0-1 mixed integer programming model is then constructed. To solve the proposed model, which is NP-hard, a developed GA is proposed to produce near-optimal solutions for large-size instances. Computational experiments on 720 instances with up to 150 jobs and 5 parallel batch-processing machines show the effectiveness and efficiency of the developed GA.

Ant colony optimization algorithm for scheduling jobs with fuzzy processing time on parallel batch machines with different capacities

We study the problem of scheduling on parallel batch processing machines with different capacities under a fuzzy environment to minimize the makespan. The jobs have non-identical sizes and fuzzy processing times. After constructing a mathematical model of the problem, we propose a fuzzy ant colony optimization (FACO) algorithm. Based on the machine capacity constraint, two candidate job lists are adopted to select the jobs for building the batches. Moreover, based on the unoccupied space of the solution, heuristic information is designed for each candidate list to guide the ants. In addition, a fuzzy local optimization algorithm is incorporated to improve the solution quality. Finally, the proposed algorithm is compared with several state-of-the-art algorithms through extensive simulated experiments and statistical tests. The comparative results indicate that the proposed algorithm can find better solutions within reasonable time than all the other compared algorithms.

Integrated production and transportation on parallel batch machines to minimize total weighted delivery time

This paper considers a production-distribution scheduling problem on parallel batch processing machines (BPMs) with multiple vehicles. In the production stage, the jobs with non-identical sizes and equal processing time are grouped into batches, which are processed on BPMs. In the distribution stage, there are vehicles with identical capacity arriving regularly to transport the batches to the customers. The objective is to minimize the total weighted delivery time of the jobs. A method of computing a lower bound is given to evaluate the proposed algorithms. To tackle this NP-hard problem, a deterministic heuristic (Algorithm H) and two hybrid meta-heuristic algorithms based on ant colony optimization (HACO, MMAS) are proposed, respectively. Through analyzing the property of the investigated problem, the heuristic information and the pheromone trails are defined. Incorporated with a local optimization strategy, the ant colony constructs the schedule first. Then, a heuristic is designed to transport the batches that have been processed. The performance of the proposed algorithms are compared with each other through testing on randomly generated problem instances. It is shown that the proposed MMAS algorithm slightly beats the HACO algorithm, which can find the better solutions than the H algorithm in a reasonable amount of time.