Single Machine Scheduling Problem Research Articles

Idle time and capacity control for a single machine scheduling problem with dynamic electricity pricing

In this paper, we develop a dynamic control algorithm for production scheduling that considers machine capacity and idle time controls and aims at satisfying time related production demand and reducing energy consumption in a unified manner. A mixed integer nonlinear programming (MINLP) model is developed to determine job arrival sequence for a machine and machine capacity while minimizing resulting costs of just-in-time production, machine repair, and energy consumption during machine idle time and nominal processing. A dynamic control algorithm based on feedback control of continuous variables is also developed to determine an energy-efficient production schedule with proper machine capacity and turn-off schedules. Energy, JIT, and maintenance costs of the proposed approach are examined using real energy and machining parameters of a HAAS VF0 milling machine. Algorithmic performance of the proposed dynamic control approach is compared to other heuristics, adaptive large neighborhood search (ALNS), and genetic algorithm (GA) with a speed optimization (SO) component. Experimental results show that the proposed algorithm improved performance by an average 10.0 ~ 93.8% and 0.52 ~ 22.9% compared to GA and ALNS with the SO module, respectively.

A Comparative Study of Scalarization Techniques on the Multi-Objective Single Machine-Scheduling Problem Under Sequence-Dependent Setup Time, Release Date and Due Date Constraints

In this paper, we considered a two-objective machine-scheduling problem under sequence-dependent setup time, release date and due date constraints. The problem is formulated as a multi-objective mixed-integer programming model. Two conflicting objectives are considered as minimization of maximum completion time (makespan) and total tardiness. Despite the most use of metaheuristics in this kind of multi-objective problems, here, we try to solve the problem by transforming the two-objectives as a single objective using scalarization techniques. Test instances are generated as proposed in the scheduling literature. The solutions are obtained using Weighted Sum Scalarization, Benson’s Method and Pascoletti−Serafini Method. In addition, a comparison of scalarization techniques using Δ performance metric is given on the considered problem instances. The obtained results are evaluated and Δ values, which were obtained for Benson’s method, are mostly better than other techniques for the generated test problems.

Approximation algorithms for the min-max regret identical parallel machine scheduling problem with outsourcing and uncertain processing time

We consider the robust (min-max regret) version of identical parallel machine scheduling problem, in which jobs may be outsourced to balance total cost against production efficiency. The total cost is measured in terms of the total completion time of jobs processed in-house and the cost of outsourcing the rest. Processing times of in-house jobs are uncertain and they are described as two types of scenarios: discrete and interval. The objective is to obtain a robust (min-max regret) decision that minimises the absolute deviation of total cost from the optimal solution under the worst-case scenario. We first prove the worst-case scenario for any feasible solution. For the interval scenario, we further prove that the maximum regret value can be obtained in polynomial time for any feasible schedule. We also prove that for any discrete scenario, the minimum total cost can be obtained in polynomial time. Since the problem with the interval scenario is strongly NP-hard, we then transform the problem into an equivalent robust single machine scheduling problem. Finally, we develop 2-approximation algorithms for the problem with discrete and interval scenarios, respectively. These results are helpful for bridging the scheduling theory and practice in identical parallel machining environments with outsourcing and uncertain processing times.

Improved Approaches to Minimize the Makespan on Single-Machine Scheduling with Periodic Preventive Maintenance Activities

This paper addresses a single-machine scheduling problem with periodic preventive maintenance activities that are predeterministic so that the machine is not available all the time, and jobs have to be processed between two consecutive maintenance periods. We propose a mixed integer programming (MIP) model and two heuristics to minimize the makespan. With more constraints in our model, the model is more efficient than the recent model of Perez-Gonzalez and Framinan , and our model could solve problems with up to fifty jobs. Two heuristic algorithms, namely, H (MW) and H (LB∗), are also proposed, in which two bin-packing policies of the minimum waste and minimum lower bound are used, respectively. Furthermore, we also proposed an improvement procedure. The results showed that the heuristic H (MW) outperformed other heuristics of the paper, indicating that the bin-packing policy of the minimum waste is more effective than well-known ones such as full batch and best fit. Additionally, all the heuristic algorithms addressed in this paper combined with the improvement procedure could achieve a similar and high quality of solutions with a very tiny increase in computational expense.

Robust parallel-batching scheduling with fuzzy deteriorating processing time and variable delivery time in smart manufacturing

Smart manufacturing is an effective way to improve the efficiency of resource utilization and reduce the response time of making joint decisions for the enterprises. Though, with the globalization of manufacturing enterprises, manufacturing optimization problems often occur in complex manufacturing systems under the deteriorating and fuzzy environment, which brings many challenges to smart manufacturing, such as the lack of coordinating scheduling strategies to guarantee the low latency requirement. This paper investigates a robust parallel-batching scheduling problem with fuzzy processing time and past-sequence-dependent delivery time. Some structural properties are first identified, and an optimal algorithm is further developed for the single-machine scheduling problem. Then, the problem is proved to be NP-hard. We thus design a hybrid Multi-Verse Optimizer-Variable Neighborhood Search algorithm to solve the investigated problem in a reasonable time. Abundant experiments of different scales are conducted to verify the performance of the proposed hybrid method with a comparison of the state-of-the-art methods. The proposed hybrid meta-heuristic shows excellent results, robustness, and computational time performance under various experiments.

Robust single machine scheduling problem with uncertain job due dates for industrial mass production

The single machine scheduling problem which involves uncertain job due dates is one of the most important issues in the real make-to-order environment. To deal with the uncertainty, this paper establishes a robust optimization model by minimizing the maximum tardiness in the worst case scenario over all jobs. Unlike the traditional stochastic programming model which requires exact distributions, our model only needs the information of due date intervals. The worst case scenario for a given sequence that belongs to a set containing only n scenarios is proved, where n is the number of jobs. Then, the model is simplified and reformulated as an equivalent mixed 0-1 integer linear programming (MILP) problem. To solve the MILP problems efficiently, a heuristic approach is proposed based on a robust dominance rule. The experimental results show that the proposed method has the advantages of robustness and high calculating efficiency, and it is feasible for large-scale problems.

Parameterized Multi-Scenario Single-Machine Scheduling Problems

We study a class of multi-scenario single-machine scheduling problems. In this class of problems, we are given a set of scenarios with each one having a different realization of job characteristics. We consider these multi-scenario problems where the scheduling criterion can be any one of the following three: The total weighted completion time, the weighted number of tardy jobs, and the weighted number of jobs completed exactly at their due-date. As all the resulting problems are NP-hard, our analysis focuses on whether any one of the problems becomes tractable when some specific natural parameters are of limited size. The analysis includes the following parameters: The number of jobs with scenario-dependent processing times, the number of jobs with scenario-dependent weights, and the number of different due-dates.

Production scheduling with autonomous and induced learning

The vast majority of scheduling research involving the learning effect only considers autonomous learning, i.e. learning by doing. Proactive investment in learning promotion, i.e. induced learning, is rarely considered. Nevertheless, induced learning is important for total production cost reduction and helping managers control the production systems, which can be interpreted as management or investment seeking to improve employees’ working efficiency. We consider in this paper scheduling models with both autonomous and induced learning. The objective is to find the optimal sequence and level of induced learning that optimise a scheduling criterion plus the investment cost. We propose polynomial-time algorithms to solve all the single-machine scheduling problems considered and the parallel-machine problem to minimise the total completion time plus the investment cost. We also propose an approximate algorithm for the parallel-machine problem to minimise the makespan plus the investment cost.

Minimizing makespan on a single machine with release dates and inventory constraints

We consider a single-machine scheduling problem with release dates and inventory constraints. Each job has a deterministic processing time and has an impact (either positive or negative) on the central inventory level. We aim to find a sequence of jobs such that the makespan is minimized while all release dates and inventory constraints are met. We show that the problem is strongly NP-hard even when the capacity of the inventory is infinite. To solve the problem, we introduce a time-indexed formulation and a sequence-based formulation, a branch-and-bound algorithm, and a dynamic-programming-based guess-and-check (GC) algorithm. From extensive computational experiments, we find that the GC algorithm outperforms all other alternatives.

A single machine scheduling problem with machine availability constraints and preventive maintenance

Considering the impact of machine availability is essential in production scheduling, especially when the target is to minimise total tardiness. In this study, a single machine scheduling problem derived from a rotor production workshop is addressed. We evaluate machine availability by its reliability, which can be restored by preventive maintenance (PM). PM with different improvement effectiveness is considered in the problem formulation. A genetic algorithm (GA) is developed to solve the problem. Emmons dominance rules are applied in the mutation processes of the GA. Computational experiments demonstrate the effectiveness and efficiency of the algorithm. Sensitivity analyses have been conducted to provide useful managerial insights for real workshop scheduling.

Single Machine Scheduling Problem with Controllable Setup and Job Processing Times and Position-Dependent Workloads

This paper concerns with a single-machine scheduling problem in which each job has a setup before being processed, the setup time and the processing time of jobs are controllable by allocating a continuously nonrenewable resource, respectively. This is different from that of [13], in which a job only has a processing time, but without a setup time. We assume that each setup and processing time of jobs has a workload that is both job and position-dependent. In the problem investigated, the total amount of resource for setup and processing times does not exceed an up bound, respectively. The objective is to determine the setup sequence, the job sequence and the resource allocation scheme to minimize makespan. We show that the problem can be solved in polynomial time and present an optimal algorithm.

Single-Machine Parallel-Batch Scheduling with Nonidentical Job Sizes and Rejection

We investigate the single-machine parallel-batch scheduling problem with nonidentical job sizes and rejection. In this problem, a set of jobs with different processing times and nonidentical sizes is given to be possibly processed on a parallel-batch processing machine. Each job is either accepted and then processed on the machine or rejected by paying its rejection penalty. Preemption is not allowed. Our task is to choose the accepted jobs and schedule them as batches on the machine to minimize the makespan of the accepted jobs plus the total rejection penalty of the rejected jobs. We provide an integer programming formulation to exactly solve our problem. Then, we propose three fast heuristic algorithms to solve the problem and evaluate their performances by using a small numerical example.

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

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

Single Machine Scheduling with Outsourcing Under Different Fill Rates or Quantity Discount Rates

In this paper, we consider two single machine scheduling problems with outsourcing under different fill rates or quantity discount rates. From the manufacturer point of view, to maintain a predefined high service level under limited production capacity, it is required that the number of outsourced jobs must belong to a given interval [Formula: see text]. We call this model “scheduling with outsourcing under different fill rates”. Meanwhile, from the outsourcers point of view, in order to encourage more outsourcing actions from the manufacturer, the outsourcers might provide an attractive discount scheme. Here, we assume that all discount rates depend only on the quantity (number) of the outsourced jobs. That is, the actual outsourcing cost equals to the original outsourcing cost multiplied by the corresponding discount rate. This model is called “scheduling with outsourcing under different quantity discount rates”. The objective is to minimize the sum of the makespan of the processed jobs and the (actual) outsourcing costs of the outsourced jobs. For the above two problems, some optimal algorithms, approximation algorithms and approximation schemes are proposed.

Common Due Window Assignment Scheduling with Proportional Linear Deterioration Effects

In this paper, single-machine scheduling problems with proportional linear deterioration effects and common due window assignment simultaneously are considered. Two different objective functions are studied, the first is to minimize the sum of the number of early jobs, number of tardy jobs and due window location and due window size, the second is to minimize the sum of the earliness cost, tardiness cost, due window location and due window size. Optimality properties for all problems are provided and polynomial time algorithms for solving these problems are given.

Approximation Algorithm for the Single Machine Scheduling Problem with Release Dates and Submodular Rejection Penalty

In this paper, we consider the single machine scheduling problem with release dates and nonmonotone submodular rejection penalty. We are given a single machine and multiple jobs with probably different release dates and processing times. For each job, it is either accepted and processed on the machine or rejected. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a nonmonotone submodular function. We design a combinatorial algorithm based on the primal-dual framework to deal with the problem, and study its property under two cases. For the general case where the release dates can be different, the proposed algorithm have an approximation ratio of 2. When all the jobs release at the same time, the proposed algorithm becomes a polynomial-time exact algorithm.

A hybrid VNS-Lagrangean heuristic framework applied on single machine scheduling problem with sequence-dependent setup times, release dates and due dates

In this work, we propose a hybrid VNS-Lagrangean heuristic applied on the single machine scheduling problem with sequence-dependent setup times, release dates, and due dates. The objective function is the minimization of the total tardiness. The proposed hybrid heuristic is a Lagrangean relaxation integrated with the variable neighborhood search (VNS). The methodology can generate strong bounds, using the information of the Lagrangean multipliers to construct and perturb feasible solutions within the VNS framework. We compare its performance with previous hybrid approaches and find that the upper bounds obtained are optimal for several cases and tight for others. The methodology presents competitive results when compared with previous related works.

Production Scheduling considering Outsourcing Options and Carrier Costs

We consider a single-machine scheduling problem with outsourcing options in an environment where the cost information of the downstream is available via some information sharing technologies. The due date is assigned to the position differently from the traditional due date. Each job can be processed in-house or outsourced. Note that, for cost saving, as many due dates as the number of outsourced jobs should be canceled. An in-house job incurs a stepwise penalty cost for tardiness, and an outsourced job incurs an outsourcing cost. Thus, the objective is to minimize the total penalty and outsourcing cost minus the total profit from cost savings. We show that the problem is weakly NP-hard and investigate some polynomially solvable cases. Due to the high complexity of the dynamic programming, we developed heuristics and verified their performance through numerical experiments.

A Single Machine Scheduling with Periodic Maintenance and Uncertain Processing Time

A single machine scheduling problem with periodic maintenance is studied in this paper. Due to many uncertainties in reality, the processing time is recognized as an uncertain variable. The aim is to minimize the makespan at a confidence level. An uncertain chance-constrained programming model is developed to delve into the impact of uncertainties on decision variables, and an algorithm for calculating the objective function is proposed. According to the theoretical analysis, a novel method named longest shortest processing time (LSPT) rule is proposed. Subsequently, an improved genetic algorithm (GA-M) combined with LSPT rule is proposed. Numerical experiments are used to verify the feasibility of this model and algorithm.

Simultaneous consideration of sequence-dependent and position-dependent setup times in single machine scheduling problems

In this work we address a single machine scheduling problems with sequence-dependent setup times, in which the setup time and the processing time may depend on the job position in the processing order. We consider two manufacturing environments. In the first one, jobs are processed automatically, then the job positions affect only on the setup times. In the second environment, the operators modify the machine settings between different types of jobs and operate it to process the jobs, then the job positions affect both setup time and processing time. In this work, we will investigate the validity of the assumption: scheduling problems render high-quality solutions that reckon with job positions just as the those reckoned without job positions. Minimising the maximum completion time and the sum of completion time of the jobs are the objective functions. To tackle this scenario, we introduce four mathematical formulations: one formulation for each combination of objective function and manufacturing environment. The validity of the models is established by the results of extensive computational experiments on the proposed models.