Parallel Batch Scheduling Research Articles

A novel carbon reduction engineering method-based deep Q-learning algorithm for energy-efficient scheduling on a single batch-processing machine in semiconductor manufacturing

The semiconductor industry is a resource-intensive sector that heavily relies on energy, water, chemicals, and raw materials. Within the semiconductor manufacturing process, the diffusion furnace, ion implantation machine, and plasma etching machine exhibit high energy demands or operate at extremely high temperatures, resulting in significant electricity consumption, which is usually carbon-intensive. To address energy conservation concerns, the industry adopts batch production technology, which allows for the simultaneous processing of multiple products. The energy-efficient parallel batch scheduling problem arises from the need to optimise product grouping and sequencing. In contrast to existing heuristics, meta-heuristics, and exact algorithms, this paper introduces the Deep Q-Network (DQN) algorithm as a novel approach to address the proposed problem. The DQN algorithm is built upon the agent’s systematic learning of scheduling rules, thereby enabling it to offer guidance for online decision-making regarding the grouping and sequencing of products. The efficacy of the algorithm is substantiated through extensive computational experiments.

A hybrid discrete differential evolution – genetic algorithm approach with a new batch formation mechanism for parallel batch scheduling considering batch delivery

Scheduling is an important decision-making problem in production planning and the resulting decisions have a direct impact on reducing waste, including energy and idle capacity. Batch scheduling problems occur in various industries from automotive to food and energy. This paper introduces the parallel p-batch scheduling problem with batch delivery, content-dependent loading/unloading times and energy-aware objective function. The problem has been motivated by a real system used for freezing products in a food processing company. A mixed-integer linear programming model (MILP) has been developed and explained through a numerical example. As it is not practical to solve large-size instances via a mathematical model, the discrete differential evolution algorithm has been improved (iDDE) and hybridised with the genetic algorithm (GA). A release-oriented vector generation procedure and a heuristic batch formation mechanism have been developed to efficiently solve the problem. The performance of the proposed approach (iDDEGA) has been compared with CPLEX, iDDE and GA through a comprehensive computational study. A case study was conducted based on real data collected from the freezing process of the company, which also verified the practical use and advantages of the proposed methodology.

Single machine parallel-batch scheduling under time-of-use electricity prices: New formulations and optimisation approaches

Electricity costs account for a relatively large proportion of total production costs in many energy-intensive manufacturing industries. And energy-efficient scheduling is getting increasing attention from researchers owing to its wide real-world applications and computational challenges. In this study, the single-machine parallel-batch scheduling problem with non-identical job sizes under time-of-use electricity prices is investigated. The objective is to minimise the total energy consumption cost such that the makespan does not exceed a given deadline. First, an improved time-slot-indexed formulation is proposed to eliminate symmetric solutions. Subsequently, a new set partition-based formulation is developed. To the best of our knowledge, there is no exact algorithm for this problem, except for some formulations solved by off-the-shelf solvers. To address this problem, a branch-and-price algorithm is developed with a novel and efficient branching rule. In addition, a column-generation-based heuristic is proposed to solve larger-scale instances. Extensive numerical experiments show that the efficiency of the branch-and-price algorithm is significantly better than that of off-the-shelf solvers, and many instances with up to 100 jobs could be solved to optimality within a 20-minute computational time limit for the first time. The column-generation-based heuristic can efficiently provide better solutions than existing heuristics (2.75% average gap in three minutes on instances with 100 jobs and 3.12% average gap in ten minutes on instances with 200 jobs).

Exact methods for the Oven Scheduling Problem

The Oven Scheduling Problem (OSP) is a new parallel batch scheduling problem that arises in the area of electronic component manufacturing. Jobs need to be scheduled to one of several ovens and may be processed simultaneously in one batch if they have compatible requirements. The scheduling of jobs must respect several constraints concerning eligibility and availability of ovens, release dates of jobs, setup times between batches as well as oven capacities. Running the ovens is highly energy-intensive and thus the main objective, besides finishing jobs on time, is to minimize the cumulative batch processing time across all ovens. This objective distinguishes the OSP from other batch processing problems which typically minimize objectives related to makespan, tardiness or lateness. We propose to solve this NP-hard scheduling problem using exact techniques and present two different modelling approaches, one based on batch positions and another on representative jobs for batches. These models are formulated as constraint programming (CP) and integer linear programming (ILP) models and implemented both in the solver-independent modeling language MiniZinc and using interval variables in CP Optimizer. An extensive experimental evaluation of our solution methods is performed on a diverse set of problem instances. We evaluate the performance of several state-of-the-art solvers on the different models and on three variants of the objective function that reflect different real-life scenarios. We show that our models can find feasible solutions for instances of realistic size, many of those being provably optimal or nearly optimal solutions.

Parallel-batch scheduling with rejection: Structural properties and approximation algorithms

In this paper, we consider a parallel-batch machine scheduling (PBMS) model that unifies several existing scheduling models in the extant literature. In our scheduling model, a given set of jobs with different release dates are scheduled onto a machine that can process multiple jobs simultaneously in a batch. However, the number of jobs in a batch is limited. Each job can be rejected subject to a job-dependent penalty cost, but the total penalty cost of the jobs rejected is required to be no greater than a given bound. The objective is to minimize the weighted sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs. We provide some important structural properties and develop fast approximation algorithms. We also present an efficient polynomial time approximation scheme (EPTAS) with near-linear-time complexity. Our results significantly improve both the time complexities and performance bounds of several existing algorithms and approximation schemes.

A Best Possible Online Algorithm For Parallel-Batch Scheduling with Kind Release Times and Job Compatibilities

A Best Possible Online Algorithm For Parallel-Batch Scheduling with Kind Release Times and Job Compatibilities

An improved algorithm on unbounded parallel-batching scheduling to minimize maximum cost and makespan

This paper studies the bicriteria problem of scheduling n jobs on a parallel-batching machine to minimize maximum cost and makespan simultaneously. A parallel-batching machine is a machine that can handle up to b jobs in a batch. The jobs in a batch start and complete respectively at the same time and the processing time of a batch is equal to the largest processing time of jobs in the batch. We consider the unbounded case. For the above bicriteria scheduling problem, we present an O(n3)-time algorithm, which improved the best known O(n4)-time algorithm, and the time complexity is the same as the special case in which maximum cost is maximum lateness. Meanwhile, our algorithm can also solve the single-criterion unbounded parallel-batching scheduling problem to minimize maximum cost in O(n3) time, which improved the best known O(n4)-time algorithm.

Online Parallel-Batch Scheduling of Learning Effect Jobs with Incompatible Job Families for Prefabricated Components

In the production scheduling of prefabricated components, we study an online [Formula: see text] parallel-batch machines scheduling model considering learning effect jobs with [Formula: see text] incompatible job families to minimize the makespan in this paper, where the capacity of batch is unbounded. Job families indicate that a job must belong to some job family and jobs of distinct job families are incapable to be executed in the same batch. The information of each job including its basic processing time [Formula: see text] and release time [Formula: see text] is unknown in advance and is revealed at the instant of its arrival. Moreover, the actual processing time of job [Formula: see text] with learning effect is [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative parameters and [Formula: see text] denotes the starting time of prefabricated job [Formula: see text], respectively. When [Formula: see text], we propose an online algorithm with a competitive ratio of [Formula: see text]. Furthermore, the performance of the online algorithm is demonstrated by numerical experiments.

Modified benders decomposition and metaheuristics for multi-machine parallel-batch scheduling and resource allocation under deterioration effect

Batch scheduling problem is a common difficulty in mass production, which includes many decisions such as grouping, batch ordering, and resource allocation. Motivated by the semiconductor burn-in test process, this paper investigates a combinatorial optimization problem considering the deterioration effect and resource allocation. The objective is to design an appropriate job sequencing and resource allocation scheme to achieve an overall minimization of makespan and resource costs. The problem, subject to parallel-batch, multi-machine, deterioration effect, and resource constraints, is NP-hard. We decompose it into a master problem and a sub-problem based on Benders Decomposition (BD). A cut filtering rule that limits the number of cuts is proposed to control the complexity of the problem. Meanwhile, some structural properties are deduced to guide the design of two algorithms combined with Variable Neighborhood Search (VNS). We conduct extensive simulation experiments on instances of different scales, and the results demonstrate that the cut filtering method can effectively reduce the solution time. In large-scale computational experiments, it is more obvious that the proposed algorithms have good convergence and robustness.

A Production Plan Considering Parallel Machines and Deteriorating Effects: Minimizing the Makespan in the Section of Steel Box Girder Processing

With the popularization of energy conservation and emission reduction, amounts of industrial production has taken energy conservation as a goal to achieve. The paper considers an online parallel-batch scheduling problem with deteriorating and incompatible families on identical machines to minimize the makespan, which minimizes the maximum energy consumption of machines. Specifically, the processing time of job [Formula: see text] is defined by an increasing function of its starting time [Formula: see text], i.e., [Formula: see text], where [Formula: see text] is the deterioration rate of job [Formula: see text]. For the problem, we propose an online algorithm with a competitive ratio of [Formula: see text], where [Formula: see text] is the largest deterioration rate in an instance. Furthermore, the paper presents a concise computational study of the numerical experiment to show that our algorithm performs very well in practice of this model.

Exact algorithms based on a constrained shortest path model for robust serial-batch and parallel-batch scheduling problems

We study robust single-machine batch scheduling problems under uncertain processing times to minimize total flow time. Two types of batches are considered: serial batch (s-batch) and parallel batch (p-batch). These problems can model many on-site production and logistics applications which involve uncertain factors such as defect rates. We first prove that a sequencing rule for the shortest nominal processing time is optimal for both s-batch and p-batch problems. We then propose polynomial-time algorithms based on the observation that the robust batch scheduling problems are reducible or partially reducible to a constrained shortest path problem through worst-case scenario analysis. We further present more efficient algorithms for the special case of uniform maximum deviation times for all jobs. The algorithms are evaluated computationally, and the results show that their performance is satisfactory on the tested instances.

Parallel batching with multi-size jobs and incompatible job families

Parallel batch scheduling has many applications in the industrial sector, like in material and chemical treatments, mold manufacturing and so on. The number of jobs that can be processed on a machine mostly depends on the shape and size of the jobs and of the machine. This work investigates the problem of batching jobs with multiple sizes and multiple incompatible families. A flow formulation of the problem is exploited to solve it through two column generation-based heuristics. First, the column generation finds the optimal solution of the continuous relaxation, then two heuristics are proposed to move from the continuous to the integer solution of the problem: one is based on the price-and-branch heuristic, the other on a variable rounding procedure. Experiments with several combinations of parameters are provided to show the impact of the number of sizes and families on computation times and quality of solutions.

A branch-and-price algorithm for robust parallel batch scheduling problem with uncertain size

PurposeIn intelligent scheduling, parallel batch processing can reasonably allocate production resources and reduce the production cost per unit product. Hence, the research on a parallel batch scheduling problem (PBSP) with uncertain job size is of great significance to realize the flexibility of product production and mass customization of personalized products.Design/methodology/approachThe authors propose a robust formulation in which the job size is defined by budget constrained support. For obtaining the robust solution of the robust PBSP, the authors propose an exact algorithm based on branch-and-price framework, where the pricing subproblem can be reduced to a robust shortest path problem with resource constraints. The robust subproblem is transformed into a deterministic mixed integer programming by duality. A series of deterministic shortest path problems with resource constraints is derived from the programming for which the authors design an efficient label-setting algorithm with a strong dominance rule.FindingsThe authors test the performance of the proposed algorithm on the extension of benchmark instances in literature and compare the infeasible rate of robust and deterministic solutions in simulated scenarios. The authors' results show the efficiency of the authors' algorithm and importance of incorporating uncertainties in the problem.Originality/valueThis work is the first to study the PBSP with uncertain size. To solve this problem, the authors design an efficient exact algorithm based on Dantzig–Wolfe decomposition. This can not only enrich the intelligent manufacturing theory related to parallel batch scheduling but also provide ideas for relevant enterprises to solve problems.

Two-agent bounded parallel-batching scheduling for minimizing maximum cost and makespan

This paper considers the bounded parallel-batching scheduling with two agents to minimize maximum cost of agent A and makespan of agent B simultaneously, in which all jobs of agent A have equal processing time, the jobs from different agents can be processed in a common batch and the cost function of each agent is only determined by its own jobs. In the paper, we present a polynomial-time algorithm to generate all Pareto optimal points for the problem and determine a corresponding Pareto optimal schedule for each Pareto optimal point.

Online Batch Scheduling of Incompatible Job Families with Variable Lookahead Interval

We consider online parallel-batch scheduling of [Formula: see text] incompatible unit-length job families with variable lookahead interval to minimize the maximum completion time, where [Formula: see text] is the number of job families which is known in advance. Incompatible job family means that a batch only contains the jobs from the same job family. At any time [Formula: see text], an online algorithm can foresee the information of the jobs arriving in the time interval [Formula: see text], where [Formula: see text] is variable. When the batch capacity [Formula: see text] and [Formula: see text], we provide a best possible online algorithm with competitive ratio [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive root of [Formula: see text]. When the batch capacity [Formula: see text] and [Formula: see text], we give an online algorithm with competitive ratio [Formula: see text] for [Formula: see text] and [Formula: see text], and prove that it is the best possible for [Formula: see text] and [Formula: see text].

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.

On lower and upper bounds for single machine parallel batch scheduling

On lower and upper bounds for single machine parallel batch scheduling

Bid construction scheme based on local search to solve serial–parallel cooperative batch scheduling problems

This study addresses cooperative batch scheduling problems based on both serial batch (s-batch) scheduling and parallel batch (p-batch) scheduling. A mixed-integer linear programming model was formulated to describe such problems and a bid construction scheme (BCS) was developed to solve them. In s-batch and p-batch scheduling, the sheet strips function as both bidders and auctioned resources, with different sheet strips forming different batches over multiple rounds of auctioning. Then, a local search operator is applied to optimize the feasible solutions. Numerical experiments were conducted on a series of randomly generated and real-world instances. The results obtained by the proposed BCS were compared with those obtained by four combinations of existing algorithms and analysed by SPSS. The proposed scheme was able to identify feasible solutions quickly and efficiently, proving its the effectiveness, stability, practicality, and suitability for addressing cooperative batch scheduling problems, especially those corresponding to large-scale instances.

Parallel batch scheduling: Impact of increasing machine capacity

Scheduling performance naturally improves with increased machine capacity, but the per-unit improvement typically decreases or even keeps unchanged with excessive capacity. We consider parallel batch scheduling on identical machines to minimize the makespan, with and without preemption. The machine capacity is the maximum number of jobs that a machine can process simultaneously. We quantitatively analyze the impact of capacity augmentation on the makespan in the form of increasing machine capacity. Considering machines costs, we need to determine the machine capacity that minimizes the weighted sum of the makespan and capacity costs. First, we obtain an upper bound of the ratio between the minimum makespans of two scheduling problems with different machine capacities. Second, noting the intractability of the scheduling problem without preemption, we analyze the upper bound of the ratio between the obtained makespans by heuristics of two scheduling problems with different machine capacities. Third, for the preemptive case, we develop a polynomial time algorithm to obtain the optimal machine capacity, and also present another polynomial time algorithm to obtain the optimal machine capacity as well as for bounding the approximation ratio of the non-preemptive case. Fourth, we design an approximation algorithm using machine capacity found in the preemptive case to yield a schedule for the non-preemptive case, and analyze the worst case performance ratio of the algorithm. This research provides new insights into performance improvement with increased machine capacity in parallel batch scheduling, and analyzes the trade-off between the makespan and capacity costs.

Research on the parallel–batch scheduling with linearly lookahead model

<p style='text-indent:20px;'>In this paper, we consider the new online scheduling model with linear lookahead intervals, which has the character that at any time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>, one can foresee the jobs that will coming in the time interval <inline-formula><tex-math id="M2">\begin{document}$ (t, \lambda t+\beta ] $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \lambda\geq1, \beta\geq 0 $\end{document}</tex-math></inline-formula>. We consider online scheduling of unit length jobs on <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula> identical parallel-batch machines under this new lookahead model to minimize the maximum flowtime and the makespan, respectively. We give some lower bounds for these problems with the batch capacity <inline-formula><tex-math id="M5">\begin{document}$ b = \infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b<\infty $\end{document}</tex-math></inline-formula>, respectively. And for the bounded model to minimize makespan, we give an online algorithm with competitive ratio <inline-formula><tex-math id="M7">\begin{document}$ 1+\alpha $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ 1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \frac{3}{2} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M10">\begin{document}$ \lambda\geq1, 0\leq\beta<1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> is the positive root of <inline-formula><tex-math id="M12">\begin{document}$ \lambda\alpha^2+(\lambda+\beta)\alpha+\beta-1 = 0 $\end{document}</tex-math></inline-formula>. The online algorithm is best possible when <inline-formula><tex-math id="M13">\begin{document}$ 1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6} $\end{document}</tex-math></inline-formula>.</p>