Machine Scheduling Problem Research Articles

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.

Single machine scheduling with due date assignment to minimize the total weighted lead time penalty and late work

We study a single machine scheduling problem with due date assignment for minimizing the total weighted lead time penalty and late work, where the lead time penalty is the excess of an assigned due date over a given lead time, and the late work is the part of a job executed after the assigned due date. For the problem with common due date assignment, we show that it is solvable in O(nlogn) time. For the problem with unrestricted due date assignment, we determine an optimal due date assignment for a given schedule, and transform the problem to an equivalent problem for minimizing the total ▪-shape function, where ▪ depicts the shape of the penalty function for a job in relation to its completion time. Furthermore, we show that the problem with unrestricted due date assignment is unary NP-hard by proving the unary NP-hardness of the equivalent problem, and study two special cases. For the case with identical lead times, we show that it is binary NP-hard by providing a dynamic programming algorithm. While for the case with identical processing times, we prove that it is solvable in O(n3) time. Finally, we study the problem of minimizing the total ▪-shape function, and show that its two special cases are binary NP-hard by proving that they are pseudo-polynomially solvable.

Sensitivity Analysis of the Unrelated Parallel Machine Scheduling Problem with Rework Processes and Machine Eligibility Restrictions

Sensitivity Analysis of the Unrelated Parallel Machine Scheduling Problem with Rework Processes and Machine Eligibility Restrictions

On some lower bounds for the permutation flowshop problem

The permutation flowshop problem with makespan objective is a classic machine scheduling problem, known to be NP-hard in the strong sense. We analyse some of the existing lower bounds for the problem, including the “job-based” and “machine-based” bounds, a bound from linear programming (LP), and a recent bound of Kumar and co-authors. We show that the Kumar et al. bound dominates the machine-based bound, but the LP bound is stronger still. On the other hand, the LP bound does not, in general, dominate the job-based bound. Based on this, we devise simple iterative procedures for strengthening the Kumar et al. and LP bounds. Computational results are encouraging. In particular, we are able to obtain improved lower bounds for the “hard, small” instances of Vallada, Ruiz and Framinan.

An optimal energy-efficient scheduling with processing speed selection and due date constraint in a single-machine environment

To relive the pressure of electricity grid during the peak period, time-of-use (TOU) pricing strategy has been implemented in industries to encourage manufacturers to transfer some processing tasks from peak periods to non-peak periods, and the due date of each job cannot be violated. Under these contexts, this paper addresses a speed-scaling single machine scheduling problem with due date constraint to minimize the total electricity cost (TEC). More precisely, the main innovative works are described as follows: (1) seven critical problem properties (including four theorems and three lemmas) based on different processing time window forms are formally derived; and (2) a property-based genetic algorithm (PGA) with hybrid initialization method is designed according to the characteristics of the studied problem. In the numerical experiments, the Taguchi method of design-of-experiment is employed to seek the optimal combination of four key parameters in PGA. Subsequently, the effectiveness and superiority of the proposed hybrid initialization method and problem properties are separately demonstrated by randomly generated 20 instances. After that, compared with other two traditional scheduling strategies, the proposed energy-efficient strategy can save at least 16% of TEC on average. Next, a relaxation coefficient (CR) is designed to measure the intrinsic link between TEC and the instance parameters (i.e. the due date and normal processing time). Finally, a real case is presented to verify the benefits of the proposed PGA algorithm and variable processing speeds, and the results show that the newly generated scheduling scheme based on the proposed PGA can reduce up to 46.36% of TEC compared with the existing company’s scheduling scheme.

제어가능한 공정시간과 납기 지연 조건을 고려한 선점적 단일 머신 일정계획 문제

In classical scheduling problems, jobs are usually assumed to be non-preemptive with constant processing times. In many practical cases, however, restrictions on preemption can be released by outsourcing and the job processing times are controllable by allocating additional resources to the job operations. In this paper, we consider a single- machine scheduling problem with preemption such that each job processing time can be reduced linearly by using additional resource. The objective is to minimize the sum of total late work and total compression cost. We show that the problem can be solved in polynomial time.

Minimizing tardiness and makespan for distributed heterogeneous unrelated parallel machine scheduling by knowledge and Pareto-based memetic algorithm

This work aims to deal with the distributed heterogeneous unrelated parallel machine scheduling problem (DHUPMSP) with minimizing total tardiness (TDD) and makespan. To solve this complex combinatorial optimization problem, this work proposed a knowledge and Pareto-based memetic algorithm (KPMA) which contains the following features: 1) four heuristic rules are designed including the shortest processing time rule, the minimum factory workload rule, the minimum machine finish time rule, and the earliest due date rule. Meanwhile, a hybrid heuristic initialization is developed to construct a population with great convergence and diversity; 2) four problem feature-based heuristic neighborhood structures are designed to increase the success rate of local search; and 3) a simple elite strategy is developed to enhance the usage of historical elite solutions. Finally, to evaluate the performance of KMPA, it is compared to five state-of-art and run on 20 instances with different scales. The results of numerical experiments show that the proposed hybrid heuristic initialization can efficiently save computation resources to improve the initialized convergence. In addition, the knowledge-based neighborhood structures can vastly accelerate exploration. Moreover, the elite strategy can efficiently improve the diversity of the final non-dominated solutions set. The proposed KPMA has better performance than the state-of-art and has a strong ability to solve DHUPMSP.

Review on unrelated parallel machine scheduling problem with additional resources

This study deals with an unrelated parallel machine scheduling problem with additional resources (UPMR). That is one of the important sub-problems in the scheduling. UPMR consists of scheduling a set of jobs on unrelated machines. In addition to that, a number of one or more additional resources are needed. UPMR is very important and its importance comes from the wealth of applications; they are applicable to engineering and scientific situations and manufacturing systems such as industrial robots, nurses, machine operators, bus drivers, tools, assembly plant machines, fixtures, pallets, electricity, mechanics, dies, automated guided vehicles, fuel, and more. The importance also comes from the concern about the limitation of resources that are dedicated for the production process. Therefore, researchers and decision makers are still working on UPMR problem to get an optimum schedule for all instances which have not been obtained to this day. The optimum schedule is able to increase the profits and decrease the costs whilst satisfying the customers’ needs. This research aims to review and discuss studies related to unrelated parallel machines and additional resources. Overall, the review demonstrates the criticality of resolving the UPMR problem. Metaheuristic techniques exhibit significant effectiveness in generating results and surpassing other algorithms. Nevertheless, continued improvement is essential to satisfy the evolving requirements of UPMR, which are subject to operational changes based on customer demand.

Review on unrelated parallel machine scheduling problem with additional resources

This study deals with an unrelated parallel machine scheduling problem with additional resources (UPMR). That is one of the important sub-problems in the scheduling. UPMR consists of scheduling a set of jobs on unrelated machines. In addition to that, a number of one or more additional resources are needed. UPMR is very important and its importance comes from the wealth of applications; they are applicable to engineering and scientific situations and manufacturing systems such as industrial robots, nurses, machine operators, bus drivers, tools, assembly plant machines, fixtures, pallets, electricity, mechanics, dies, automated guided vehicles, fuel, and more. The importance also comes from the concern about the limitation of resources that are dedicated for the production process. Therefore, researchers and decision makers are still working on UPMR problem to get an optimum schedule for all instances which have not been obtained to this day. The optimum schedule is able to increase the profits and decrease the costs whilst satisfying the customers’ needs. This research aims to review and discuss studies related to unrelated parallel machines and additional resources. Overall, the review demonstrates the criticality of resolving the UPMR problem. Metaheuristic techniques exhibit significant effectiveness in generating results and surpassing other algorithms. Nevertheless, continued improvement is essential to satisfy the evolving requirements of UPMR, which are subject to operational changes based on customer demand.

Approximation algorithms for scheduling single batch machine with incompatible deteriorating jobs

Motivated by the soaking process under separate heating mode in iron and steel enterprises, we study the parallel batch machine scheduling problem with incompatible deteriorating jobs. The objective is to minimize makespan. A soaking furnace can be seen as a parallel batch processing machine. In order to avoid the thermal stress caused by excessive temperature difference, initial temperature is needed for the ingot before processing. With the increasing of waiting time, the ingot temperature decreases and the soaking time increases. This property is called deterioration. Setup time is needed between incompatible jobs. We show that if jobs have the same sizes, an optimal solution can be found withinO(nlogn) time. If jobs have identical processing times, the problem is proved to be NP-hard in the strong sense. We propose an approximate algorithm whose absolute and asymptotic worst-case ratios are less than 2 and 11/9, respectively. When the jobs have arbitrary sizes and arbitrary processing times, the model is also NP-hard in the strong sense. An approximate algorithm with an absolute and asymptotic worst-case ratio less than 2 is proposed. The time complexity isO(nlogn).

Algorithms for single machine scheduling problem with release dates and submodular penalties

Algorithms for single machine scheduling problem with release dates and submodular penalties

Unrelated parallel machine scheduling models with machine availability and eligibility constraints

In this paper, we address the unrelated parallel machine scheduling problem with calendar, which consists of a generic intermittent availability scheme with non-regular active and inactive time windows in sequence. The problem additionally considers machine eligibility and release date constraints, as well as the possibility of pausing or not the processing of the jobs between successive active time windows. This problem is quite frequent and relevant in several practical settings in which it is necessary to schedule parallel resources with different calendars, such as different weekly and daily shifts, non-regular not worked days, holidays, and preventive maintenance periods, among others, while considering the resources eligibility. We present novel mixed integer linear programming models for this problem with the aim of minimizing the makespan. Basic formulations of these models are presented as well as tight lower bounds for the objective, including enhanced formulations. To the best of our knowledge, there are no other works that tackled this problem and that proposed optimization models with the level of detail and generality here considered. The models presented are evaluated using a state-of-the-art general-purpose solver, and the computational tests are performed on a variety of instances. The results validate the models and show that they are can handle instances in realistic situations where the problem occurs. The results also show that the enhanced formulations are able to find several optimal solutions, and that within acceptable computational times they are able to sharply reduce the optimality gaps when compared to the more basic formulations.

Two-Stage Genetic Algorithm for Scheduling Stochastic Unrelated Parallel Machines in a Just-in-Time Manufacturing Context

This paper considers a stochastic parallel machine scheduling problem in a just-in-time manufacturing context, in which its processing time can be described by a gamma or log-normal distribution. In order to obtain a high-performance schedule in a reasonable time, this work proposes a two-stage genetic algorithm with optimal computing budget allocation (OCBA) and improved Monte-Carlo Policy Evaluation (MCPE). In it, a genetic algorithm is selected as a main optimizer. An OCBA-based approach is developed to improve search efficiency, which is designed for two scenarios in a just-in-time manufacturing context. Different from most prior OCBA studies, this work considers that the stochastic processing time of jobs does not obey normal distribution. It extends the application area of OCBA by laying a theoretical foundation. A parameter control scheme based on MCPE is proposed, which aims to balance the global and local search in GA. To further enhance the efficiency and effectiveness of the proposed method, a two-stage framework is constructed. In the first stage, the performance is estimated roughly aiming at locating satisfactory solution regions. In the second stage, OCBA is incorporated to provide the reliable evaluation of excellent individuals. The theoretic interpretation of the proposed OCBA, and the convergence analysis results of the proposed method are presented. Various simulation results with benchmark and randomly generated cases validate that the proposed algorithm is more efficient and effective than several existing optimization algorithms. Note to Practitioners—A parallel machine scheduling problem under stochastic processing time is usually solved via meta-heuristic algorithms. However, their computational efficiency requires substantial improvement, especially for a stochastic optimization case that requires Monte Carlo sampling to estimate the actual objective function values in a precise manner. Most of them are parameter-sensitive, and choosing their proper parameters is highly challenging. For the first thorny issue, we develop an OCBA-based approach for determining the optimal numbers of simulations according to both prior knowledge and simulation results. In order to select proper control parameters of the proposed algorithm iteratively, we introduce a parameter control scheme based on MCPE. The combination of a meta-heuristic algorithm, OCBA and MCPE makes it possible to find high-quality solutions for the concerned scheduling problems in a short time. Theoretic analysis and numerical simulation results suggest that the proposed framework is valid and efficient. Hence, it can be readily applicable to practical systems, e.g., semiconductor manufacturing.

Ensembles of priority rules to solve one machine scheduling problem in real-time

Priority rules are one of the most common and popular approaches to real-time scheduling. Over the last decades, several methods have been developed to generate rules automatically. In addition, it has been shown that combining rules into ensembles is better than using a single rule in many cases. In this paper, we analyze different ways to create and use ensembles previously developed through genetic programming. In our study, we classify ensembles as either collaborative or coordinated, depending on how the rules are used. In the first case, all the rules contribute to the creation of the same solution, while in the second case, each rule works independently on its own solution, and the best of them is selected as the solution of the ensemble. We found that each method has its own strengths and weaknesses, which leads us to use them in combination. Based on this hypothesis, we developed new methods to design and combine collaborative and coordinated ensembles and evaluated these methods for the One Machine Scheduling Problem with time-varying capacity and minimization of total tardiness. The results of the experimental study provided interesting insights into the use of ensembles and showed that our proposals outperform previous methods.

Mixed-Integer Programming vs. Constraint Programming for Shop Scheduling Problems: New Results and Outlook

Constraint programming (CP) has been recently in the spotlight after new CP-based procedures have been incorporated into state-of-the-art solvers, most notably the CP Optimizer from IBM. Classical CP solvers were only capable of guaranteeing the optimality of a solution, but they could not provide bounds for the integer feasible solutions found if interrupted prematurely due to, say, time limits. New versions, however, provide bounds and optimality guarantees, effectively making CP a viable alternative to more traditional mixed-integer programming (MIP) models and solvers. We capitalize on these developments and conduct a computational evaluation of MIP and CP models on 12 select scheduling problems. 1 We carefully chose these 12 problems to represent a wide variety of scheduling problems that occur in different service and manufacturing settings. We also consider basic and well-studied simplified problems. These scheduling settings range from pure sequencing (e.g., flow shop and open shop) or joint assignment-sequencing (e.g., distributed flow shop and hybrid flow shop) to pure assignment (i.e., parallel machine) scheduling problems. We present MIP and CP models for each variant of these problems and evaluate their performance over 17 relevant and standard benchmarks that we identified in the literature. The computational campaign encompasses almost 6,623 experiments and evaluates the MIP and CP models along five dimensions of problem characteristics, objective function, decision variables, input parameters, and quality of bounds. We establish the areas in which each one of these models performs well and recognize their conceivable reasons. The obtained results indicate that CP sets new limits concerning the maximum problem size that can be solved using off-the-shelf exact techniques. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1287 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0326 ) at ( http://dx.doi.org/10.5281/zenodo.7541223 ).

A Single Machine Group Scheduling Problem with Due Date Assignment and Position-Dependent Costs

In this paper, we study a single machine scheduling problem with due date assignment under a group technology environment. The due dates of jobs are assignable according to the following three methods: the common (CON) due date assignment, the slack (SLK) due date assignment, and the different (DIF) due date assignment. Our objective is to determine a group sequence and job sequences within all groups, together with a due date assignment, to minimize a total cost function including earliness, tardiness, due date assignment and flow time costs, in which all unit costs are different and position-dependent. We present a unified [Formula: see text] time algorithm to solve the problem for all the three due date assignment methods.

A two-stage RNN-based deep reinforcement learning approach for solving the parallel machine scheduling problem with due dates and family setups

As an essential scheduling problem with several practical applications, the parallel machine scheduling problem (PMSP) with family setups constraints is difficult to solve and proven to be NP-hard. To this end, we present a deep reinforcement learning (DRL) approach to solve a PMSP considering family setups, aiming at minimizing the total tardiness. The PMSP is first modeled as a Markov decision process, where we design a novel variable-length representation of states and actions, so that the DRL agent can calculate a comprehensive priority for each job at each decision time point and then select the next job directly according to these priorities. Meanwhile, the variable-length state matrix and action vector enable the trained agent to solve instances of any scales. To handle the variable-length sequence and simultaneously ensure the calculated priority is a global priority among all jobs, we employ a recurrent neural network, particular gated recurrent unit, to approximate the policy of the agent. The agent is trained based on Proximal Policy Optimization algorithm. Moreover, we develop a two-stage training strategy to enhance the training efficiency. In the numerical experiments, we first train the agent on a given instance and then employ it to solve instances with much larger scales. The experimental results demonstrate the strong generalization capability of the trained agent and the comparison with three dispatching rules and two metaheuristics further validates the superiority of this agent.

A shuffled frog-leaping algorithm with Q-learning for unrelated parallel machine scheduling with additional resource and learning effect

Unrelated parallel machine scheduling problem (UPMSP) with additional resources and UPMSP with learning effect have attracted some attention; however, UPMSP with additional resources and learning effect is seldom studied and meta-heuristics for UPMSP hardly possess reinforcement learning as new optimization mechanism. In this study, a shuffled frog-leaping algorithm with Q-learning (QSFLA) is presented to solve UPMSP with one additional resource and learning effect. A new solution presentation is presented. Two populations are obtained by division. A Q-learning algorithm is constructed to dynamically decide search operator and search times. It has 12 states depicted by population quality evaluation, four actions defined as search operators, a new reward function and a new action selection. Extensive experiments are conducted. Computational results demonstrate that QSFLA has promising advantages for the considered UPMSP.

Hybridizing genetic algorithm and single-based metaheuristics to solve unrelated parallel machine scheduling problem with scarce resources

<span lang="EN-US">This paper focuses on solving unrelated parallel machine scheduling with resource constraints (UPMR). There are </span><em><span lang="EN-US">j</span></em><span lang="EN-US"> jobs, and each job needs to be processed on one of the machines aim at minimizing the </span><em><span lang="EN-US">makespan</span></em><span lang="EN-US">. Besides the dependence of the machine, the processing time of any job depends on the usage of a rare renewable resource. A certain number of those resources R</span><em><span lang="EN-US">max</span></em><span lang="EN-US"> can be disseminated to jobs for the purpose of processing them at any time, and each job j needs units of resources (</span><em><span lang="EN-US">rjm</span></em><span lang="EN-US">) when processing in machine m. When more resources are assigned to a job, the job processing time minimizes. However, the number of resources available is limited, and this makes the problem difficult to solve for a good quality solution. Genetic algorithm shows promising results in solving UPMR. However, genetic algorithm suffers from premature convergence, which could hinder the resulting quality. Therefore, the work hybridizes genetic algorithm with a single-based metaheuristics (SBH) to handle the premature convergence in the genetic algorithm with the aim to escape from the local optima and improve the solution quality further. The single-based metaheuristics replaces the mutation in the genetic algorithm. The evaluation of the algorithm performance was conducted through extensive experiments. </span>

Single machine scheduling problems involving job-dependent step-deterioration dates and job rejection

Single machine scheduling problems involving job-dependent step-deterioration dates and job rejection