Parallel-machine Problem Research Articles

Scheduling parallel batch processing machines: A case study in the semiconductor industry

This paper presents a mathematical programming-based solution approach for the scheduling problem of batch-processing parallel machines with eligibility constraints. A case study has been presented in the semiconductor industry, where the ovens are scheduled for the underfill cure operation of products. The case includes constraints, such as oven-product eligibility restrictions, loading constraints for the batching of products for ovens, daily production requirements, and oven capacity constraints. In this study, we also assess the difference between creating batches of a single product type or different product types to be allocated to the ovens. The case study results have shown that the proposed models, in comparison to the current situation, increase the occupancy rate of ovens. The execution of the models aids the company in gaining visibility on the scheduling of ovens and successfully managing the production plan and order commitment. The proposed models have been effective and supportive of the semiconductor company.

An application of mathematical programming to a real case of the unrelated parallel machine problem

An application of mathematical programming to a real case of the unrelated parallel machine problem

Parallel batch processing machines scheduling for SLM additive manufacturing in cloud manufacturing

For the scheduling problem of parallel batch processing machines for Selective Laser Melting (SLM) additive manufacturing in cloud manufacturing environments, an Improved Discrete Artificial Bee Colony algorithm(IABC) is proposed by considering the service time of the product as well as introducing the mathematical model of the problem. Firstly, a population initialization strategy is introduced to improve the quality of initial solutions and accelerate the convergence of the population.Secondly, through three neighborhood search operators, the transition from the original nectar source to the new nectar source is achieved for employed bees and following bees. Furthermore, a local search strategy and a modified abandonment pattern are proposed to increase population diversity and prevent the algorithm from falling into local optima.Finally, the performance of the algorithm is analyzed, and the IABC algorithm was applied to 36 examples. The effectiveness and advantages of IABC in solving the studied problem are verified through a large number of simulation experiments.

A novel exact formulation for parallel machine scheduling problems

Machine scheduling is one of the most studied problems due to its technical challenges and prevalence in real life. In the literature, continuous- and discrete-time formulations are the two most known formulations for scheduling problems. However, continuous-time formulations often suffer from weak linear relaxations, while discrete-time formulations struggle with large numbers of variables. In contrast, the bucket-indexed formulation is an alternative that mitigates both issues by working with partial time discretization. We propose a mixed-integer linear programming model based on a bucket-indexed formulation to solve a nonpreemptive scheduling problem of identical parallel machines considering release dates, deadlines, precedence, eligibility, and machine availability constraints. We evaluate the proposed formulation against real-world instances comprising more than 400 jobs and 100 machines, comparing its performance against equivalent continuous- and discrete-time formulations. Remarkably, our formulation can be solved to optimality for all instances, outperforming both continuous- and discrete-time formulations.

A Metaheuristic Framework for Energy-Intensive Industries With Batch Processing Machines

Batch processing machines, which operate multiple jobs at a time, are commonly used in energy-intensive industries. A significant amount of energy can be saved in such industries using production scheduling as an approach to enhance efficiency. This study deals with an energy-aware scheduling problem for parallel batch processing machines with incompatible families and job release times. In such an environment, a machine may need to wait until all the jobs in the next batch become ready. During waiting time, a machine can be switched <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> or kept <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> standby for more energy-efficient scheduling. We first present a mixed-integer linear programming (MILP) model to solve the problem. However, the presented MILP model can only solve small problem instances. We therefore propose an energy-efficient tabu search (ETS) algorithm for solving larger problem instances. The proposed solution framework incorporates multiple neighborhood methods for efficient exploration of the search space. An energy-related heuristic is also integrated into the ETS for minimizing energy consumption during the waiting time. The performance of our proposed ETS algorithm is validated by comparing it with CPLEX for small problem instances and with two other heuristic algorithms for larger problem instances. The contribution of different components in ETS is also established in our experimental studies. The proposed solution framework is expected to bring many benefits in energy-intensive industries both economically and environmentally.

Joint scheduling of AGVs and parallel machines in an automated electrode foil production factory

In recent years, the automated guided vehicle (AGV) scheduling problem and parallel machine scheduling problem (PMSP) in workshops have become two hot topics in the field of operational research. In practical production, the two problems are closely related and coupled, and both play important roles in manufacturing systems. However, in the literature, the two problems are often addressed separately, which decreases the efficiency of manufacturing systems to a great degree. Therefore, this paper studies the joint scheduling problem of AGV and parallel machines in a production process of automatic electrode foil. Taking makespan as the optimization objective, a mixed-integer linear programming model is established and verified by the GUROBI solver. A discrete gray wolf optimization algorithm is proposed to solve the AGV-PMSP. An improved NEH heuristic is used to propose the quality of the initial solution. Six neighborhood operators are used to improve the exploration capabilities of the proposed algorithm. A theorem is proved to improve computational efficiency. The performance of the proposed algorithm is verified using comprehensive simulation experiments.

Energy-saving and noise-reducing integrated task allocation model for machining systems and its application

Abstract In this paper, firstly, based on the application model of optimal scheduling of machining system for green manufacturing, two application models of the energy-saving scheduling model and energy-saving and noise-reducing scheduling model in a multi-model framework are combined, and the resource environment coefficient matrix of the two application models is established as well as the solution process is studied with the parallel machine problem. Then the system's architecture is constructed, and its basic operation flow, functional modules, etc., are designed and conceived. The application of the system is studied in conjunction with a gear machining workshop of a machine tool factory, and the machining system's energy and noise reduction performance is verified based on experiments. The results show that the energy consumption of the machining system is reduced by 0.514 kW-h by machining only the above six gear parts with a small difference in the maximum machining completion time and that the spindle speed has the most significant effect on the machine tool machining noise at a significance level α of 0.05. The analysis of this study verifies that the energy-saving and noise-reducing scheduling arrangement method can reduce the system machining energy consumption and noise, which is important for green manufacturing.

Quantum annealing solution for the unrelated parallel machine scheduling with priorities and delay of task switching on machines

Quantum computing has emerged in recent years as an alternative to classical computing, which could improve the latter in solving some types of problems. One of the quantum programming models, Adiabatic Quantum Computing, has been successfully used to solve problems such as graph partitioning, traffic routing, and task scheduling. In this paper, the focus is on the scheduling of the problem of unrelated parallel machines, where the processing time of tasks on any of the available processing elements is known. Moreover, the proposed model is extended in two relevant aspects for this kind of problem: the existence of some degree of priority of tasks, and the introduction of a delay or penalty every time a processing unit or machine changes the type of task that executes.In all cases, the problem is expressed as Quadratic Unconstrained Binary Optimization, which can be subsequently solved using quantum annealers. The quantum nonlinear programming framework discussed in this work consists of three steps: quadratic approximation of cost function, a binary representation of parameter space, and solving the resulting Quadratic Unconstrained Binary Optimization on the quantum annealer platform D-Wave. One of the novelties in tackling this problem is the compaction of the model bearing in mind the repetitions of each task, to allow solving larger scheduling problems with the quantum resources available in the experimentation platform. An estimation of the number of qubits required in relation to the scheduling parameters is analyzed. The models have been implemented on the D-Wave platform and validated with respect to other traditional methods. Furthermore, the proposed extensions to consider priorities and to switch the delay of tasks have been analyzed using a case study.

A Genetic-Algorithm-Based Approach for Optimizing Tool Utilization and Makespan in FMS Scheduling

This paper proposes a genetic algorithm approach to solve the identical parallel machines problem with tooling constraints in job shop flexible manufacturing systems (JS-FMSs) with the consideration of tool wear. The approach takes into account the residual useful life of tools and allocates a set of jobs with specific processing times and tooling requirements on identical parallel machines. Two metrics are introduced to evaluate the scheduling decisions and optimize the scheduling process, with the competitive goal of maximizing tool utilization and minimizing production makespan. The proposed approach searches for a set of optimal solutions on the Pareto front that offers the best possible balance between these two objectives, achieving optimal local performance in terms of both makespan and tool utilization. The approach is implemented with a customized genetic algorithm and validated on a real case study from a company operating in the aerospace sector, which confirms its effectiveness in increasing tool utilization and reducing the makespan. The results show that the proposed approach has significant practical implications for the manufacturing industry, particularly in the production of high-value materials such as those in the aerospace sector that require costly tools. This paper contributes to the operational research community by providing advanced scheduling algorithms that can optimize both the makespan and the tool utilization concurrently, improving production efficiency and maintaining competitiveness in the manufacturing industry.

Quick dispatching-rules-based solution for the two parallel machines problem under mold constraints

Quick dispatching-rules-based solution for the two parallel machines problem under mold constraints

Hybrid Flow Shop with Setup Times Scheduling Problem

The two-stage hybrid flow shop problem under setup times is addressed in this paper. This problem is NP-Hard. on the other hand, the studied problem is modeling different real-life applications especially in manufacturing and high performance-computing. Tackling this kind of problem requires the development of adapted algorithms. In this context, a metaheuristic using the genetic algorithm and three heuristics are proposed in this paper. These approximate solutions are using the optimal solution of the parallel machines under release and delivery times. Indeed, these solutions are iterative procedures focusing each time on a particular stage where a parallel machines problem is called to be solved. The general solution is then a concatenation of all the solutions in each stage. In addition, three lower bounds based on the relaxation method are provided. These lower bounds present a means to evaluate the efficiency of the developed algorithms throughout the measurement of the relative gap. An experimental result is discussed to evaluate the performance of the developed algorithms. In total, 8960 instances are implemented and tested to show the results given by the proposed lower bounds and heuristics. Several indicators are given to compare between algorithms. The results illustrated in this paper show the performance of the developed algorithms in terms of gap and running time.

An Improved Arc Flow Model with Enhanced Bounds for Minimizing the Makespan in Identical Parallel Machine Scheduling

In this paper, an identical parallel machine problem was considered with the objective of minimizing the makespan. This problem is NP-hard in the strong sense. A mathematical formulation based on an improved arc flow model with enhanced bounds was proposed. A variable neighborhood search algorithm was proposed to obtain an upper bound. Three lower bounds from the literature were utilized in the improved arc flow model to improve the efficiency of the mathematical formulation. In addition, a graph compression technique was proposed to reduce the size of the graph. As a consequence, the improved arc flow model was compared with an arc flow model from the literature. The computational results on benchmark instances showed that the improved arc flow model outperformed the literature arc flow model at finding optimal solutions for 99.97% of the benchmark instances, with the overall percentage of the reduction in time reaching 87%.

Scheduling unrelated parallel machine problem with multi-mode processing times and batch delivery cost

Scheduling unrelated parallel machine problem with multi-mode processing times and batch delivery cost

An approximation algorithm for the two identical parallel machine problem under machine availability constraints

ABSTRACT This study addresses the scheduling problem of two identical parallel machines with the objective of minimizing the total completion time under the machine availability constraints. To the best of our knowledge, this study is the first to develop a fully polynomial-time approximation scheme (FPTAS), a solution method which has been neglected in past studies, to solve the studied problem. The FPTAS, which is based on a dynamic programming algorithm is developed by applying a trimming-the-state-space approach. Theoretical proofs of the error bound and the time complexity for the proposed FPTAS are also provided. The computational results indicate that the proposed FPTAS performs more efficiently than a dynamic programming algorithm in terms of both run time and problem size. The error bound of the FPTAS is demonstrated to be within the pre-specified error bound.

Column generation for minimizing total completion time in a parallel-batching environment

This paper deals with the 1|{p-text {batch}, s_jle b}|sum C_j scheduling problem, where jobs are scheduled in batches on a single machine in order to minimize the total completion time. A size is given for each job, such that the total size of each batch cannot exceed a fixed capacity b. A graph-based model is proposed for computing a very effective lower bound based on linear programming; the model, with an exponential number of variables, is solved by column generation and embedded into both a heuristic price and branch algorithm and an exact branch and price algorithm. The same model is able to handle parallel-machine problems like Pm|{p-text {batch}, s_jle b}|sum C_j very efficiently. Computational results show that the new lower bound strongly dominates the bounds currently available in the literature, and the proposed heuristic algorithm is able to achieve high-quality solutions on large problems in a reasonable computation time. For the single-machine case, the exact branch and price algorithm is able to solve all the tested instances with 30 jobs and a good amount of 40-job examples.

Exact makespan minimization of unrelated parallel machines

We study methods for the exact solution of the unrelated parallel machine problem with makespan minimization, generally denoted as R||C max . Our original application arises from the automotive assembly process where tasks needs to be distributed among several robots. This involves the solutions of several R||C max instances, which proved hard for a MILP solver since the makespan objective induces weak LP relaxation bounds. To improve these bounds and to enable the solution of larger instances, we propose a branch–and–bound method based on a Lagrangian relaxation of the assignment constraints. For this relaxation we derive a criterion for variable fixing and prove the zero duality gap property for the case of two parallel machines. Our computational studies indicate that the proposed algorithm is competitive with state-of-the-art methods on different types of instances. Moreover, the impact of each proposed feature is analysed.

Integer programming formulations in sequencing with total earliness and tardiness penalties, arbitrary due dates, and no idle time: A concise review and extension

Scheduling with arbitrary due dates and no idle time permitted between jobs with the objective of minimizing the total earliness and tardiness has a variety of industry applications. We review mathematical programming formulations for single and parallel machine problems with fixed and controllable processing and setup times. Weaknesses of the formulations are identified, corrections and/or improvements are provided, and further research directions are suggested.

Integrated Scheduling of Tasks and Preventive Maintenance Periods in a Parallel Machine Environment With Single Robot Server

In this study, the objective of minimizing makespan has been considered for a scheduling problem of identical parallel machines with a single server and unavailability constraints. The unavailability constraints correspond to preventive maintenance periods. In this study, the jobs and the maintenance periods are scheduled simultaneously. This scheduling problem has a wide range of potential application areas in the manufacturing environment. In addition, the studied problem is a challenging one from theoretical point of view, due to its NP-Hardness. To conduct the study, a lower bound ( LB ) for the problem, and three metaheuristics namely Simulated Annealing ( SA ), Tabu Search ( TS ), and Genetic Algorithm ( GA ) have been proposed. The best parameters settings of the proposed algorithms were conducted using pilot runs with a Taguchi design. The algorithms performance has been assessed by using a set of test problems generated randomly. These test problems are based on a literature benchmark. The size of the instances, or number of jobs, were up to 500. Along with the performance analysis of the proposed algorithms, the effect of varying processing times and unavailability periods on the performance of the proposed algorithms is studied. The present work provides strong evidence of the efficiency and the performance of the proposed algorithms.

Energy-awareness scheduling of unrelated parallel machine problems with multiple resource constraints

The environmental and economic pressures caused by energy consumption arouse energy-saving consciousness of the manufacturing industry. To this end, the paper integrates energy-awareness into the research of the unrelated parallel machines scheduling problem with multiple auxiliary resources which is typical in the photolithography process of wafer fabrication. With a comprehensive consideration of jobs requiring different processing demands, setup times, different ready times, resource constraints and energy consumption, a scheduling model with a multi-objective function of minimising the total weighted completion time and total energy consumption of the system is developed. On the basis of the model, a modified multi-objective artificial immune algorithm integrated with non-domination sorting strategy is put forward to crack the problem. Furthermore, in order to improve the performance of the proposed algorithm, clone operators, neighbourhood search operators, elite-preservation operators are applied to the algorithm. Finally, the experimental results and analysis validate that the presented algorithm is efficient and effective.

Feature-Extraction-Based Iterated Algorithms to Solve the Unrelated Parallel Machine Problem With Periodic Maintenance Activities

This paper considers an unrelated parallel machine problem with job release times and maintenance activities, in which machines have to periodically undergo maintenance since the status of the machines will be deteriorated by job-induced dirt. The problem is inspired by a wet station for cleaning operations in a semiconductor manufacturing process. The objective is to minimize the makespan. Since the considered problem is proven to be NP-hard, obtaining optimal solutions is almost impossible in a reasonable computational time when the problem becomes large. We develop specific feature-extraction procedures to recognize important information in a job sequence and linkage encoding (LE) procedures to generate new job sequences. The two above procedures are embedded into an iterated algorithm, called a feature-extraction-based iterated algorithm (FEBIA), to obtain optimal or better solutions for the considered problem. To examine the performance of the FEBIA, the FEBIA is compared with two population-based algorithms, the particle swarm optimization (PSO) algorithm and the genetic algorithm (GA), using many test data. The results reveal that the proposed FEBIA perform better than the two population-based algorithms, demonstrating the potential of the FEBIA to solve the unrelated parallel machine problem with periodic maintenance and job release times.