Machine Scheduling Problem Research Articles

New mathematical model for machine scheduling with variable capacity over time

ABSTRACT In this study, machine scheduling with variable capacity over time (SVCap) is investigated. The machine capacity is the maximum number of jobs that a machine can process at a time which can be either fixed or variable over time. In common machine scheduling problems, it is assumed that one machine can process one job at a time. However, in variable machine capacity, multiple jobs can be processed on a machine simultaneously. Unlike the current research, a mathematical formulation is not developed yet for solving this problem. In order to solve the problem, a novel mixed integer linear programming (MILP) is proposed. In addition, the SVCap is regarded as a special type of resource-constrained project scheduling problem (RCPSP). Thus, the discrete-time (DT) formulation is generalized to solve the SVCap. In these formulations, the total tardiness is minimized as the objective function. Proposed models are implemented on an irrigation scheduling problem in which water resources are allocated to each plot of farmland. The computational performances of proposed formulations are evaluated on problem instances with different sizes. Results show that the proposed formulations solved all problem instances. The results demonstrate that the proposed MILP formulation is more efficient than the generalized DT formulation in both solution quality and runtimes.

Cooperative game theory based coordinated scheduling of production and transportation of parallel machines with deterioration effects

Abstract Cooperative game theory is used to study the production and transportation scheduling problem of parallel machines with deterioration effects. Jobs with an initial scheduling order are processed on parallel machines and then transported to customers. Penalties are incurred for early or late completion if the jobs are not completed on the due dates. Considering that jobs can rearrange the scheduling order in the coalition through cooperation to bring about cost savings, the cooperative game model is constructed by considering the jobs as players and taking the maximum cost savings of the coalition as the characteristic function. It is proven that when jobs have the same normal processing time and deterioration rate, the cooperative game has a non-empty core, and a core allocation can be obtained from the β-rule allocation method. A Dual Replay Buffer Double DQN (DRB-DDQN) algorithm is designed to get the optimal schedule of the coalition. Experiments demonstrate the effectiveness of the DRB-DDQN and β-rule allocation method in solving the game scheduling problem.

A multi-objective artificial bee colony algorithm for single machine scheduling with family setup under TOU tariffs

Time-of-use (TOU) electricity tariffs have been widely implemented in the manufacturing industry in many countries. This paper investigates a single machine scheduling problem involving incompatible job families with sequence-dependent setup times to minimise total electricity cost and total tardiness simultaneously. To tackle this problem, we propose a multi-objective artificial bee colony (MABC) algorithm. Utilising the dominance properties of the problem, we develop tailored heuristics aimed at improving the quality of initial food sources, and design multi-directional neighbourhood structures to explore desirable neighbour solutions along each objective direction. We construct a novel fitness function that not only considers Pareto rank but also incorporates the hypervolume contribution indicator to identify the promising solution space. Moreover, local integer programming is embedded into the MABC algorithm to intensify the search towards Pareto solutions. The experimental results indicate that the MABC algorithm performs significantly better than NSGA-II, SPEA2, and MOEA/D algorithms.

Approximation Schemes for Parallel Machine Scheduling to Maximize Total Weighted Early Work With a Common Due Date

ABSTRACTIn this article, we study a parallel machine scheduling problem with a common due date to maximize total weighted early work. We present the first EPTAS (efficient polynomial time approximation scheme) for this scheduling problem. In the EPTAS, we first schedule the large jobs with rounded processing times via an integer programming formulation and a network‐flow‐based algorithm, followed by further scheduling the small jobs via a greedy method. For the special case in which the number of machines is fixed, we design an FPTAS (fully polynomial time approximation scheme).

Optimizing energy efficiency in unrelated parallel machine scheduling problem through reinforcement learning

The industrial sector plays a significant role in global energy consumption and greenhouse gas emissions. To reduce this environmental impact, it's crucial to implement energy-efficient manufacturing systems that utilize sustainable materials and optimize energy usage. This can lead to benefits such as reduced carbon footprints and cost savings.In recent years, metaheuristic approaches have been focused on minimizing energy consumption within the Unrelated Parallel Machine Scheduling Problem (UPMSP). Traditional methods often overlook complex factors like release dates, due dates, and job setup times. This research introduces a novel algorithm that integrates reinforcement learning (RL) with a genetic algorithm (GA) to address this gap.The proposed RLGA algorithm, rooted in the dynamic field of evolutionary reinforcement learning, breaks down policies into smaller components to isolate essential parameters for problem-solving. Through comprehensive analysis, hyperparameters that influence optimal results are identified, facilitating automated hyperparameter selection and optimization. The expert system takes into account problem characteristics such as machine or job saturation, job overlap, and the maximum values of target variables, allowing instances to be grouped into clusters. These clusters are solved using a genetic algorithm with varying combinations of mutation and crossover hyperparameters. The most suitable approach for each cluster is determined by analyzing the results, and this configuration of hyperparameters is applied iteratively to optimize the solution search.The effectiveness of RLGA is evaluated across benchmark instances with different complexities, machine sets, jobs, and constraints. Comprehensive comparisons against existing methods highlight the superior performance and efficiency of RLGA in optimizing energy use and solution quality. Experimental results show that RLGA outperforms well-known solvers like CPO, CPLEX, OR-tools, and Gecode, making it a promising approach for optimizing energy-efficient manufacturing systems.

A new branch-and-cut approach for integrated planning in additive manufacturing

In recent years, there has been considerable interest in the transformative potential of additive manufacturing (AM) since it allows for producing highly customizable and complex components while reducing lead times and costs. The rise of AM for traditional and new business models enforces the need for efficient planning procedures for AM facilities. In this area, the assignment and sequencing of components to be built by an AM machine, also called a 3D printer, is a complex challenge combining two combinatorial problems: The first decision involves the grouping of parts into production batches, akin to the well-known bin packing problem. Subsequently, the second problem pertains to the scheduling of these batches onto the available machines, which corresponds to a parallel machine scheduling problem. For minimizing makespan, this paper proposes a new branch-and-cut algorithm for integrated planning for unrelated parallel machines. The algorithm is based on combinatorial Benders decomposition: The scheduling problem is considered in the master problem, while the feasibility of an obtained solution with respect to the packing problem is checked in the sub-problem. Current state-of-the-art techniques are extended to solve the orthogonal packing with rotation and used to speed up the solution of the sub-problem. Extensive computational tests on existing and new benchmark instances show the algorithm’s superior performance, improving the makespan by 18.7% on average, with improvements reaching up to 97.6% for large problems compared to an existing integrated mixed-integer programming model.

Parallel Machine Scheduling Problems with Learning and Deterioration Effects

Parallel Machine Scheduling Problems with Learning and Deterioration Effects

Novel mathematical formulations for parallel-batching processing machine scheduling problems

We study mathematical formulations for batch-processing machine scheduling problems (BPMPs), which are the challenging issues in the machine scheduling literature where machines are capable of processing a batch of jobs simultaneously if jobs with non-identical sizes can be packed in a capacitated machine. In this paper, we tackle single- and parallel-machine BPMPs, and other interesting problem variants that aim at minimizing the makespan. We develop novel formulations along with valid inequalities and an algorithm framework that makes use of dual information and bounding techniques to achieve efficiency when instances are intractable. Extensive computational experiments on benchmark instances show that our approaches achieve state-of-the-art results and prove the optimality of intractable instances in the literature.

Lower and upper bounds for scheduling a real-life assembly problem with precedences and resource constraints

In this work, we consider a real-life scheduling problem involving the production of off-road vehicles using a three-level assembly process, subject to precedence, machines, and resource constraints. This problem shares multiple characteristics with other scheduling problems such as the Parallel Machine Scheduling and Flexible Flow Shop Problems. However, it also comprises less investigated aspects such as a specific job precedence structure resulting in a directed rooted in-tree and a global resource constraint limiting the number of simultaneously active machines. We present a straightforward adaptation of a time-indexed mathematical formulation to the problem and introduce a new lower-bounding procedure. To solve the problem, we propose constructive heuristics based on classical priority rules and adaptations of job sequencing heuristics. We also propose CORE, a specialized approach tailored to leverage the problem structure. All the algorithms are extensively evaluated on two benchmark sets, various shop floor configurations, and eight real-life scenarios. Our results reveal the general effectiveness of job sequencing approaches and demonstrate the overall superiority of CORE.

Minimizing total completion time and makespan for a multi-scenario bi-criteria parallel machine scheduling problem

Multi-criteria scheduling problems under uncertainty remain a relatively unexplored topic in theoretical computer science despite substantial practical interests. This work studies a bi-objective identical parallel machine scheduling problem under uncertainty, in which the first objective is to minimize the total completion time, and the second is to minimize the makespan. Especially a job’s processing time is assumed to be represented by a polynomial function with respect to scenario u∈U, where U⊂R+ is an interval containing an infinite number of scenarios.In this work, we are looking for a compact and complete description of the set of possibly optimal solutions, along with their objective function values, over the set of scenarios or an approximation with a performance guarantee. First, to better understand the characteristics of the studied problem, we consider two single-objective problems: parallel machine scheduling problem under uncertainty with the total completion time and makespan criterion, respectively. For the problem with the total completion time criterion, we demonstrate that the set of possibly optimal schedules can be found in polynomial time. In contrast, for the problem with the makespan criterion, we provide a (1+ϵ)-approximation algorithm. For the bi-objective problem, we provide a 2-approximation algorithm for any number of parallel machines and a (1+ϵ)-approximation algorithm where a fixed number of parallel machines is considered.

A new composite heuristic to minimize the total tardiness for the single machine scheduling problem with variable and flexible maintenance

Inspired by a real-world maintenance/job scheduling issue coming from the semiconductor industry, the present paper proposes a new heuristic algorithm structure for the single machine scheduling T-problem with flexible and variable maintenance, job release dates and sequence dependence setup times. Considering the typical short-term production planning needs, a single maintenance problem has to be scheduled within a certain time interval, along with a set of jobs so as to minimize the total tardiness. A twofold contribution emerges from the present paper. First, four mixed-integer linear programming models are developed for the problem at hand and compared in terms of time to convergence and computational complexity. Second, a novel heuristic algorithm, which has been configured into three distinct variants, has been compared with 17 alternative heuristics from the relevant literature based on a comprehensive experimental campaign. The numerical results allow the selection of the most suitable MILP model and confirm the effectiveness of the proposed heuristic approach.

Single machine scheduling with the total weighted late work and rejection cost

AbstractWe study a single machine scheduling problem with the total weighted late work and the total rejection cost. The late work of a job is the part of this job executed after its due date, and the rejection cost of a job is the fee of rejecting to process it. We consider a Pareto scheduling and two restricted scheduling problems. The Pareto scheduling problem aims to find all non‐dominated values of the total weighted late work and the total rejection cost. The restricted scheduling problems are dedicated to minimizing the total weighted late work with the total rejection cost not greater than a given threshold, and minimizing the total rejection cost with the total weighted late work not greater than a given threshold, respectively. For the Pareto scheduling problem, we prove that it is binary NP‐hard by providing a pseudo‐polynomial time algorithm, and give a fully polynomial time approximation scheme (FPTAS). For the restricted scheduling problems, we prove that there are no FPTASes unless , which answers an open problem. Moreover, we develop relaxed FPTASes for these two restricted scheduling problems.

A transformer-based deep reinforcement learning approach for dynamic parallel machine scheduling problem with family setups

A transformer-based deep reinforcement learning approach for dynamic parallel machine scheduling problem with family setups

The Parallel Machine Scheduling Problem with Different Speeds and Release Times in the Ore Hauling Operation

The Parallel Machine Scheduling Problem with Different Speeds and Release Times in the Ore Hauling Operation

Parallel machine scheduling with a total energy consumption limitation for minimizing total completion time

This article investigates an energy-efficient, identical parallel machine scheduling problem. The objective is to minimize the total completion time while total energy costs do not exceed a given limitation. First, a reasonable range of total energy cost limits and the NP-hardness of this problem are analysed. Then, a mixed-integer programming model is presented. Afterwards, an improved simulated annealing (ISA) algorithm is devised. The ISA algorithm incorporates four effective neighbourhood operators, a method for adaptive selection of operators, and an initial feasible solution generation method. Comparison with genetic algorithms (GAs) and simplified swarm optimization (SSO) algorithms verifies the effectiveness and efficiency of the ISA algorithm. Results demonstrate that the ISA algorithm can provide near-optimal or better solutions than CPLEX ® in small- and medium-scale instances. Experimental results on large-scale instances show that ISA significantly outperforms GAs and SSOs. ISA, in particular, can find the best values for all instances of 500 jobs except one.

A distributionally robust approach for the parallel machine scheduling problem with optional machines and job tardiness

This paper investigates a parallel machine scheduling problem with uncertain job processing time, where the job tardiness and optional machines are considered. To address the factor of energy saving, only a subset of all available machines are turned on, which is referred to as not-all-machine (NAM). To depict the uncertain processing time, a mean–mean absolute deviation (MAD) ambiguity set is utilized, and the cost of job tardiness is minimized under the worst-case distribution scenario over the ambiguity set. After building a distributionally robust optimization (DRO) model, theoretical bounds of the optimal number of machines are obtained. Since the model is not computationally scalable, an upper bound on its inner minimization problem is employed, and a mixed integer linear programming (MILP) approximation is obtained based on McCormick inequalities. For the DRO model, tailored speedup techniques are employed, significantly enhancing the computational performance. To evaluate the validity of the proposed DRO model, we compare it with its stochastic programming (SP) counterpart under various parameter settings. Numerical experiments demonstrate that the DRO model exhibits strong performance in the worst-case scenarios. As the problem size increases, the DRO model casts clear advantages over the SP model in terms of computational efficiency and reliability. It is observed that the performance of the DRO model is more stable than that of the nominal sequence, especially with loose due dates. Furthermore, the out-of-sample performance under various decision making preferences shed new lights into the trade-off between energy saving and production efficiency.

Advanced constraint programming formulations for additive manufacturing machine scheduling problems

In comparison with traditional subtractive manufacturing techniques, additive manufacturing (AM) enables fabricating complex parts through a layer-by-layer process. AM makes it possible to produce one-piece and lightweight functional products, which are traditionally made from several parts. This paper introduces constraint programming (CP) models to minimise makespan in single, parallel identical and parallel unrelated AM machine scheduling environments for selective laser melting. Alternative CP formulations are explored to improve efficiency. The proposed CP model significantly benefits from the introduction of interval variables to replace binary assignment variables, and pre-definitions to narrow the search space, resulting in increased search performance. A computational study has been conducted to compare the performance of our proposed CP model with both a mixed-integer programming and a genetic algorithm from existing literature, evaluating improvements made to its search capability. Computational results indicate that the proposed CP model can obtain high-quality solutions in a timely manner even for several large-size instances.

An incremental learning approach to dynamic parallel machine scheduling with sequence-dependent setups and machine eligibility restrictions

Minimizing the tardiness of a parallel machine scheduling problem has been actively studied in modern manufacturing systems. In particular, dynamic parallel machine scheduling problems (DPMSPs) have gained much attention since rescheduling is required to address unpredictable events such as unexpected job arrivals and machine breakdowns, which usually occur in real-world manufacturing systems. To deal with such unpredictable events, many researchers have employed deep supervised learning (SL). However, it is still challenging to solve a DPMSP since SL requires a large number of high-quality schedules for the training neural networks. In this paper, we propose an incremental learning-based scheduling method (ILS) in which neural networks (NNs) are periodically trained by utilizing schedules built from the updated NNs in previous training intervals. Furthermore, to perform training without reducing solution space within a dynamic environment, the proposed method is designed to consider the dependency between consecutive allocations on a machine for the sum of setup time and tardiness. To verify the effectiveness of the proposed method, extensive experiments are carried out in static and dynamic scheduling problems. The experiment results demonstrate that ILS outperforms the existing methods such as dispatching rules, metaheuristics, and DRL-based methods. Additionally, we show that the proposed input features are effective in appropriately selecting job-machine pairs for assigning jobs on eligible machines through deep Shapley additive explanations analysis.

Just-in-time scheduling of unrelated parallel machines with family setups and soft time window constraints

ABSTRACT This study addresses an unrelated parallel machine scheduling problem with family setups and soft time windows, which includes considerations for machine eligibility. The objective is to minimize the total weighted tardiness and earliness, and to maximize the number of Just-in-Time jobs. Four different metaheuristic algorithms are compared to determine which type – single-point (simulated annealing), population-based evolutionary (artificial immune system, genetic algorithm), or swarm intelligence algorithms (ant colony optimization) – is more effective in solving this problem. Two strategies, namely penalty and repair, are proposed to handle precedence constraints. Experimental results demonstrate that ant colony optimization with local search which employs the repair strategy, as a swarm intelligence-based algorithm outperforms its rivals in terms of objective function values and robustness. The performance of this algorithm has shown an improvement in objective function values ranging from 2.12% to 8.52% on average, with a relative percentage deviation of 10.35.

A restless bandit approach for capacitated condition based maintenance scheduling

AbstractThis paper considers the maintenance scheduling problem of multiple non-identical machines deteriorating over time. The deterioration gradually decreases a machine’s performance, which results in revenue losses due to lower output quality. The maintenance cost is dependent on the degradation state, and the number of maintenance activities that can be carried out simultaneously is restricted by the number of maintenance workers. Our main goal is to propose a heuristic with low complexity that consistently produces solutions close to the optimal strategy for problems of real size. We cast the problem as a restless bandit problem and propose an index based heuristic (Whittle’s index policy) which can be computed efficiently. We also provide a lower bound that can be computed by linear programming. We numerically compare the performance of the index heuristic with alternative policies. In addition to achieving superior performance over failure-based and threshold policies, Whittle’s policy numerically converges to our lower bound when the number of machines is moderately high and/or maintenance workload is high.