Identical Parallel Machines Research Articles

A Comparative Study of Evolutionary and Swarm Intelligence Algorithms for Job Scheduling on Identical Parallel Machines

In parallel computing systems, job scheduling plays a crucial role in enhancing system efficiency and minimizing the makespan. In recent years, evolutionary and swarm intelligence algorithms have gained prominence as effective approaches for solving combinatorial optimization problems. In the present work, we have considered genetic algorithm (GA) for evolutionary algorithms and particle swarm optimization (PSO) for swarm intelligence algorithms. Evolutionary algorithms (EA) and swarm intelligence algorithms (SIA) have shown promising results in solving job scheduling challenges. In this study, we collate the performance of EA and SIA approaches for job scheduling on parallel machines. We use different benchmark instances to evaluate the algorithms' makespan and computational time performance. The results show that SIA algorithms outperform EA algorithms regarding makespan and computational time for all benchmark instances. Furthermore, the study provides insights into the strengths and weaknesses of EA and SIA algorithms for job scheduling on parallel machines. Our findings provide useful insights for researchers and practitioners interested in applying optimization techniques to solve job scheduling problems on parallel machines.

The LPT heuristic for minimizing total load on a proportionate openshop

We study the problem of minimizing total load on a proportionate openshop. The problem is proved to be NP-hard. A simple LPT (Longest Processing Time first)-based heuristic is proposed, and a bound on the worst-case relative error is introduced: 76−16m (where m is the number of machines). The proposed bound is smaller than the classical bound on the relative error of LPT when minimizing makespan on parallel identical machines. The algorithm is tested numerically and is shown to produce very close-to-optimal schedules.

Tailored Iterated Greedy metaheuristic for a scheduling problem in metal 3D printing

This article contributes to the additive manufacturing-based production planning literature by developing a Mixed-Integer Linear Programming (MILP) formulation for the Identical Parallel 3D-Printing Machines Scheduling Problem considering batching, multiple build platforms of restricted sizes, and sequence-independent setup times. Besides, a customized metaheuristic, named the Tailored Iterated Greedy (TIG) Algorithm is developed to solve the new optimization problem. TIG's performance is evaluated through extensive numerical analysis and using a new testbed. It is shown that the customized computational mechanisms improve the optimization performance; statistical analysis is supportive of the significance of the resulting improvements. The developed mathematical model and optimization algorithm can be considered the basis for future developments in the optimization literature of additive manufacturing.

Integrated production scheduling and vehicle routing problem with energy efficient strategies: Mathematical formulation and metaheuristic algorithms

This paper addresses integrated production and distribution scheduling problem in which orders/jobs are undergone a single operation on any one of the identical machines in parallel and upon the completion of the production they are distributed to destined customers by limited number of vehicles. Customers located in dispersed regions place their orders with predetermined demand size and time windows. On production side, machines operate under discrete speed modes, low of which requires less energy cost or vice versa. On distribution side, energy consumed by a vehicle varies depending on the size of load on it. Therefore, objective is minimizing the sum of the weighted cost emanating from early and tardy deliveries plus production and distribution costs. Operational decisions for (i) production are to determine the allocation of jobs to the machines and sequence of jobs on any machine as well as speed mode of each machine for a particular job. As for (ii) distribution: We need to decide vehicle assignment to specific subset of consolidated jobs and the sequence of customer visitation for each vehicle. We develop a formulation for the problem at hand involving parallel machine scheduling and vehicle routing to obtain solutions to optimality. Not surprisingly, however, CPLEX only provides optimum solution for all instances with customer number up to and 6, for which reason we present two metaheuristics, namely Memetic Algorithm (MA) and Iterated Local Search (ILS) for practical sized instances. Computational results indicate that ILS yields better solutions in shorter times as compared to its counterpart.

Reinforcement learning for energy-efficient control of parallel and identical machines

Nowadays, the growing interest in industry for enhancing the sustainability of manufacturing processes is becoming a major trend. Energy consumption can be lowered by controlling machine states with energy-efficient control policies that switch off/on the device. Recent studies have shown that Reinforcement Learning algorithms can effectively control manufacturing systems without the requirement of prior knowledge about system parameters. This is a significant factor since full information on system dynamics is difficult to obtain in real-world applications. This work proposes a new Reinforcement Learning-based algorithm to apply energy-efficient control strategies to a single workstation consisting of identical parallel machines. The model goal is to achieve the optimum trade-off between system productivity and energy demand without relying on full knowledge of the system dynamics. Numerical experiments confirm effectiveness, applicability, and generality of the proposed approach, even when applied to a real-world industrial system from the automotive sector.

Scheduling multi-staged jobs on parallel identical machines and a central server with sequence-dependent setup times: An application to an automated kitchen

We address the problem of scheduling multi-staged jobs in a production environment with parallel identical machines and a central server with sequence-dependent setup times motivated by a real-life application in a robotic automated kitchen. The proposed model addresses a makespan minimization offline optimization problem that arises in the context of mise-en-place preparation. We present two mathematical formulations, one with a machine index and one without, as well as lower bounds. We also propose an effective general variable neighborhood search algorithm based on heuristics which aims to improve the incumbent by eliminating idle times. Through a series of experiments on random instances, we demonstrate the effectiveness of the proposed method under various combinations of instance parameters. On instances with known optimal solutions, the metaheuristic produces solutions with an average gap of 0.31% in very short computation times. On large-size instances, the metaheuristic produces solutions with an average gap of 6.04% when measured against a known lower bound. We also present a real-life case study provided by our industrial partner and we investigate how to maximise the output of the studied automated production system.

Unsupervised parallel machines scheduling with tool switches

This paper addresses the problem of scheduling jobs on identical parallel machines with tool switches in a high-mix, low-volume manufacturing environment. Inspired by the initiatives on “lights-out factory” at our industry partner, our problem setting involves several complex features. For example, we consider unsupervised production hours (e.g., night shifts where operators are not available) in which tool switches cannot occur. Moreover, motivated by current practice, tool switches in our problem setting cause costs instead of delays. Also, a subset of jobs is prioritized to be completed within a scheduling horizon, and a job may consist of ordered operations due to reentry to machines. The objective is to maximize the profit generated by the manufacturing system, which is composed of revenue generated by the finished operations minus tool switching costs and penalty costs of unfinished priority jobs. The decisions involve assigning operations to machines, sequencing these operations, and determining a tool-switching plan. A mix-integer linear programming model is first formulated. We then propose a genetic algorithm to solve industry-size problem instances, in which tailored crossover and mutation mechanisms are introduced. We illustrate the performance of the proposed GA with industry case studies using real-world data. We also make the anonymized data set publicly available. Computational experiments reveal that approximately 26% profit improvement can be achieved by using the proposed GA instead of the current way of scheduling at our industry partner. Moreover, we find that the proposed GA brings higher benefits when the duration of the unsupervised shifts gets longer, and there is high pressure on prioritizing jobs in the schedule.

Identical Parallel Machine Scheduling Considering Workload Smoothness Index

Workload balance is significant in the manufacturing industry. However, on the one hand, some existing specific criteria cannot achieve the minimization workload imbalance of parallel machines. On the other hand, there are few algorithms in existing studies that can effectively solve the parallel machine scheduling problem with the objective of minimizing workload imbalance. Inspired by this, we investigate an identical parallel machine scheduling problem with the objective of the minimum workload smoothness index. We first establish a mathematical model for the considered problem and then linearize its objective function. We prove the NP-hardness of the problem by reducing the PARTITION problem to it, and we provide both the upper bound and lower bound of the studied problem. An efficient genetic algorithm and an improved list scheduling algorithm are also proposed to efficiently address the considered problem. The numerical results demonstrate the effectiveness of the proposed methods.

Scheduling coupled tasks on parallel identical machines

In many applications in the context of patient appointment scheduling there are recurring tasks with fixed delays between them. These tasks are commonly referred to as coupled-task jobs. In the coupled-task settings, each job consists of two tasks whereby the second task must start processing after an exact time lag following the completion of the first task. In this paper, we introduce the problem of scheduling a set of coupled-task jobs on parallel identical machines with the objective function of minimizing the makespan. We study the computational complexity of the general problem, as well as its special cases. We prove that the majority of these problems are (strongly) NP-hard. Nonetheless, we provide the optimal scheduling policy for two settings consisting of identical jobs. An important result of our work includes showing that the existence of a (2-ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2-\\varepsilon )$$\\end{document}-approximation algorithm for the problem implies P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P=NP$$\\end{document}. The latter result improves a recently proposed bound for the open-shop counterpart as well.

Robust identical parallel machine scheduling with two-stage time-of-use tariff and not-all-machine option

Time-of-use (TOU) tariff has been implemented in the manufacturing industry to improve energy efficiency by regulating the electricity imbalance between supply and demand. Besides, not-all-machine (NAM) option is another way of energy-saving by using only a subset of all the available machines. This study investigates a robust identical parallel machine scheduling problem with a two-stage TOU tariff and NAM option. Only interval bounds on job processing times are known. The problem is first formulated into a min–max regret model to maximise the robustness. Based on problem properties, both an iterative relaxation-based exact algorithm and a memetic differential evolution-based heuristic are developed to solve the problem. Computational experiments on 240 randomly generated instances with up to 20 jobs are conducted to evaluate the performance of the developed methods. Besides, 900 large-sized randomly generated instances with up to 150 jobs are tested for sensitivity analysis and to identify managerial insights for achieving energy-efficient schedules.

Scheduling Jobs with a Limited Waiting Time Constraint on a Hybrid Flowshop

In this paper, we address a two-stage hybrid flowshop scheduling problem with identical parallel machines in each stage. The problem assumes that the queue (Q)-time for each job, which represents the waiting time to be processed in the current stage, must be limited to a predetermined threshold due to quality concerns for the final product. This problem is motivated by one that occurs in the real field, especially in the diffusion workstation of a semiconductor fabrication. Our objective is to minimize the makespan of the jobs while considering product quality. To achieve this goal, we formulated mathematical programming, developed two dominance properties for this problem, and proposed three heuristics with the suggested dominance properties to solve the considered problem. We conducted simulation experiments to evaluate the performance of the proposed approaches using randomly generated problem instances that are created to closely resemble real production scenarios, and the results demonstrate their superiority over existing methods. Furthermore, we applied the proposed methods in a real-world setting within the semiconductor fabrication industry, where they have exhibited better performance compared to the dispatching rules commonly used in practical applications. These findings validate the effectiveness and applicability of our proposed methodologies in real-world scenarios.

An FPTAS for scheduling with resource constraints

We study a basic scheduling problem with resource constraints: A number of jobs need to be scheduled on two parallel identical machines with the objective of minimizing the makespan, subject to the constraint that jobs may require a unit of one of the given renewable resources during their execution. For this NP-hard problem, we develop a fully polynomial-time approximation scheme (FPTAS). Our FPTAS makes a novel use of existing algorithms for the subset-sum problem and the open shop scheduling problem.

Optimal Scheduling of Rainwater Collection Vehicles: Mixed Integer Programming and Genetic Algorithms

Due to climate change, some areas in the world witnessed higher levels of heavy rain than the capacity of the wastewater system of the streets. Therefore, water tankers are used for the dewatering process to take the extra rainwater from the streets to keep a smooth flow of vehicles and to use the water in agriculture and industry. Water is taken to a water treatment plant. Performing the dewatering process as fast as possible, especially in crowded streets, was ignored by researchers. In this study, at first, the problem was solved using two mixed integer programming (MIP) models. A new variant of identical parallel machine scheduling with job splitting is proposed for the first time, where one or at most two tankers can work at the same flood location at the same time. This is performed in the second model. However, the first model considers dividing the dewatering processes into two phases, where the first one, which is more urgent, is to reduce the amount of floodwater. The second one is for dewatering the rest of the water. Then two genetic algorithms (GAs) were used to solve faster the two MIP models, which are NP-hard problems. At first, the MIP and GA models were applied to small-sized problems. Then GA was used for large practical data sets. Results showed that for small problems, MIP and GA gave optimal solutions in a reasonable number of iterations, while for larger problems, good solutions were obtained in a reasonable number of iterations.

Exact and heuristic solution approaches for energy-efficient identical parallel machine scheduling with time-of-use costs

Nowadays, energy-efficient scheduling has assumed a key role in ensuring the sustainability of manufacturing processes. In this context, we focus on the bi-objective problem of scheduling a set of jobs on identical parallel machines to simultaneously minimize the maximum completion time and the total energy consumption over a time horizon partitioned into a set of discrete slots. The energy costs are determined by a time-of-use pricing scheme, which plays a crucial role in regulating energy demand and flattening its peaks. First, we uncover a symmetry-breaking property that characterizes the structure of the solution space of the problem. As a consequence, we provide a novel, compact mixed-integer linear programming formulation at the core of an efficient exact solution algorithm. A thorough experimental campaign shows that the use of the novel mathematical programming formulation enables the solution of larger-scale instances and entails a reduction in the computational times as compared to the formulation already available in the literature. Furthermore, we propose a new heuristic that improves the state-of-the-art in terms of required computational effort and quality of solutions. Such a heuristic outperforms the existing heuristics for the problem and is also capable of speeding up the exact solution algorithm when used for its initialization. Finally, we introduce a novel dynamic programming algorithm that is able to compute the optimal timing of the jobs scheduled on each machine to further improve the performance of the new heuristic.

Bag-of-Tasks Scheduling with Rejection in Large Computing Systems

We are given a set of [Formula: see text] identical parallel machines and a set of [Formula: see text] jobs in large computing systems, where each job [Formula: see text] consists of a bag of [Formula: see text] identical tasks with a processing time [Formula: see text], and has a rejection penalty [Formula: see text]. Job [Formula: see text] is either accepted in which case all the [Formula: see text] tasks must be processed by one of the machines, or rejected which incurs a rejection penalty [Formula: see text]. The problem of bag-of-tasks scheduling with rejection is to find a feasible schedule, so as to minimize the makespan plus the total rejection penalty of all rejected jobs. In this paper, we present a polynomial time approximation scheme.

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.

Semi-Online Scheduling on Two Identical Parallel Machines with Initial-Lookahead Information

This work investigates a new semi-online scheduling problem with lookahead. We focus on job scheduling on two identical parallel machines, where deterministic online algorithms only know the information of [Formula: see text] initial jobs (i.e., the initial-lookahead information), while the following jobs still arrive one-by-one in an over-list fashion. We consider makespan minimization as the objective. The study aims at revealing the value of knowing [Formula: see text] initial jobs, which are used to improve the competitive performance of those online algorithms without such initial-lookahead information. We provide the following findings: (1) For the scenario where the [Formula: see text] initial jobs are all the largest jobs with length [Formula: see text], we prove that the classical LIST algorithm is optimal with competitive ratio [Formula: see text]; (2) For the scenario where the total length of these [Formula: see text] jobs is at least [Formula: see text], we show that any online algorithm has a competitive ratio at least 3/2, implying that the initial-lookahead knowledge is powerless since there exists a 3/2-competitive online algorithm without such information; (3) For the scenario where the total length of these [Formula: see text] jobs is at least [Formula: see text] ([Formula: see text]), we propose an online algorithm, named as LPT-LIST, with competitive ratio of [Formula: see text], implying that the initial-lookahead information indeed helps to improve the competitiveness of those online algorithms lacking such information.

Novel Heuristic Algorithm for Flexible Job Shop Scheduling based on the Longest Processing Time Rules to Minimize Makespan

This research examined the scheduling of jobs with multiple stages unto identical parallel machines to minimize the makespan. The work is motivated by a Flexible Manufacturing System case that produces various parts and has multiple machining centers. Early research towards this system proposed a stage-by-stage independent scheduling, resulting in a non-optimal solution. This study aimed to create a better solution for the system by developing a novel heuristic algorithm based on the classical longest processing time algorithm and simultaneously considering processing time for all stages when deciding the job sequencing and job-machine allocation. The algorithm is defined as Modified LPT for Multiple Identical Machine with Multi-process Capability (M-LPT MIMMPC). We performed a numerical experiment to assess the algorithm's performance by incorporating various cases. We concluded that the resulting makespans are always better than LPT's theoretical bound for parallel machine scheduling. In some cases, it successfully gave an optimal value. Although the experiment scope was still limited, the algorithm showed promising performance results.

재투입이 존재하는 유연흐름공정에서 주문의 총 흐름시간을 최소화하기 위한 일정계획 방법론

This study focuses on a scheduling problem in a flexible flow shop, where there are serial stages, each with identical parallel machines. We suggest heuristic algorithms for the problem to minimize the total flow time of a given set of orders. Each order is composed of multiple lots and is processed on any parallel machines at each stage. This shop has reentrant flows since products for specific orders should visit the processing stages twice. We suggest algorithms of two types. The algorithms of the first type are three-phase algorithms, and the second type's algorithm is one-phase. In the first phase of three-phase algorithms, an initial sequence is obtained using priority dispatching rules, and then a construction algorithm is used to obtain sequences for each stage in the second phase. Finally, in the third phase, lots that visit the stages for the second time are scheduled using priority dispatching rules. Computational experiments are performed on randomly generated test problems to evaluate the performance of the suggested algorithms. Results show that the order-based algorithms perform better than the lot-based algorithms and an algorithm used in practice.

Lot scheduling involving completion time problems on identical parallel machines

Lot scheduling involving completion time problems on identical parallel machines