Parallel Machine Scheduling Problem Research Articles

A Hybrid Imperialist Competitive Algorithm for the Distributed Unrelated Parallel Machines Scheduling Problem

In this paper, the distributed unrelated parallel machines scheduling problem (DUPMSP) is studied and a hybrid imperialist competitive algorithm (HICA) is proposed to minimize total tardiness. All empires were categorized into three types: the strongest empire, the weakest empire, and other empires; the diversified assimilation was implemented by using different search operator in the different types of empires, and a novel imperialist competition was implemented among all empires except the strongest one. The knowledge-based local search was embedded. Extensive experiments were conducted to compare the HICA with other algorithms from the literature. The computational results demonstrated that new strategies were effective and the HICA is a promising approach to solving the DUPMSP.

Hybrid meta-heuristics for the unrelated parallel machine scheduling problem with setup times

This study focuses on an unrelated parallel machine scheduling problem with machine and job sequence-dependent setup times (UPMST) aiming to minimise the makespan. Recently, adaptive large neighbourhood search with learning automata (LA-ALNS) has been employed to tackle the UPMST with excellent performance. However, its search efficiency is heavily affected by re-visiting the recent solutions, which is known as the short-term cycle. Moreover, it remains challenging to balance the computational cost and quality of the solutions. To address these challenges, we propose a hybrid meta-heuristic method based on LA-ALNS and tabu search (LA-ALNS-TS). In the proposed algorithm, LA-ALNS is used to guide the exploration and TS is used to alleviate the short-term cycle experienced during the search process. In addition, we design two local search operators to achieve a balanced trade-off between the search efficiency and quality of the solutions. A dynamic perturbation scheme is proposed to improve the solution and help the algorithm escape from local optima. Comprehensive experiments were conducted to assess the performance of the algorithm on two exhaustive benchmarks. Experimental results demonstrate the effectiveness and efficiency of the proposed hybrid meta-heuristic method.

Hybrid tabu search algorithm for unrelated parallel machine scheduling in semiconductor fabs with setup times, job release, and expired times

This research is motivated by a scheduling problem arising in the ion implantation process of wafer fabrication. The ion implementation scheduling problem is modeled as an unrelated parallel machine scheduling (UPMS) problem with sequence-dependent setup times that are subject to job release time and expiration time of allowing a job to be processed on a specific machine, defined as: R|rj,eij,STsd|Cmax. The objective is first to maximize the number of processed jobs, then minimize the maximum completion time (makespan), and finally minimize the maximum completion times of the non-bottleneck machines. A mixed-integer programming (MIP) model is proposed as a solution approach and adopts a hybrid tabu search (TS) algorithm to acquire approximate feasible solutions. The MIP model has two phases and attempts to achieve the first two objectives. The hybrid TS algorithm has three phases and attempts to achieve all three objectives. In a real setting, computational results demonstrate that the maximum number of processed jobs can be acquired within a short time utilizing the hybrid TS algorithm (average 8 s). By comparing the two approaches, the TS outperforms the MIP model regarding solution quality and computational time for the second objective, minimizing the makespan. Furthermore, the third phase of the hybrid TS algorithm shows the effectiveness further to enhance the utilization of the ion implantation equipment.

Some Scheduling Problems With Job Rejection And A Learning Effect

AbstractIn this paper, we examined single and parallel machine scheduling problems with a learning effect and job rejection simultaneously. In real life, job processing times decrease when there is a learning effect. In some cases, producers cannot process all the jobs and pay the penalty cost for these jobs that they do not process. In our study, learning effect and job rejection are considered at the same time. We examined four different objective functions. Our objectives for single-machine scheduling problems are makespan and rejection cost minimization, total completion time and rejection cost minimization and total absolute deviation of completion times (TADC) and rejection cost minimization. Our objective for parallel machines is makespan and rejection cost minimization. The problems are solved by mathematical models, and four different algorithms are proposed for the problems. From these algorithms, the same results are obtained with single-machine makespan and rejection cost minimization, parallel machine makespan and rejection cost minimization and total completion time and rejection cost minimization. The accuracy for these models is obtained as 100%. The proposed algorithm for TADC and rejection cost minimization yielded close-to-optimal results. Mathematical model and algorithm results for 10 jobs, 20 jobs and 30 jobs are compared and the results are presented. The obtained solutions are obtained in polynomial time.

Approximation algorithm for the parallel-machine scheduling problem with release dates and submodular rejection penalties

Approximation algorithm for the parallel-machine scheduling problem with release dates and submodular rejection penalties

Modeling and solving a real world machine scheduling problem with due windows and processing set restrictions

Motivated by a real-world problem situation faced by a large manufacturing company in the last step of their production process, we model and solve an offline, deterministic, non-preemptive, identical, parallel machine scheduling problem with processing set restrictions, availability constraints, release dates, bounded tardiness, blocking constraints and due windows. The objective is to minimize the total tardiness while penalizing the assignment of jobs to larger than necessary machines. We develop a mixed integer linear program and we propose a heuristic solution approach relying on the well known large neighborhood search paradigm which relies on tailored destroy and repair operators, embedded into a simulated annealing framework. The proposed algorithm is tested on artificial instances as well as on real-world data provided by the company and compared to optimal solutions obtained by means of CPLEX. The developed tool is now used for decision support by the company partner.

A New Approach Based on the Learning Effect for Sequence‐Dependent Parallel Machine Scheduling Problem under Uncertainty

Production system design has lots of restrictions and complex assumptions that cause difficulty in decision making. One of the most important of them is the complexity of the relationship between man and machine. In this regard, operator learning is recognized as an effective element in completing tasks in the production system. In this research, a mathematical model for scheduling the parallel machines in terms of job degradation and operator learning is presented. As one of the most important assumptions, the sequence‐dependent setup time is of concern. In other words, jobs are processed sequentially, and there is a sequence‐dependent setup time. Moreover, the processing time and delivery due date are considered uncertain, and a fuzzy conversion method is used to deal with this uncertainty. The proposed mathematical model is a multiobjective one and tries to minimize speed and completion time. In order to optimize this mathematical model, the genetic algorithm (GA) and variable neighborhood search (VNS) algorithms have been used. A new hybrid algorithm has also been developed for this problem. The results show that the hybrid algorithm can provide more substantial results than classical algorithms. Moreover, it is revealed that a large percentage of Pareto solutions in the proposed algorithm have a generation time of more than 80% of the algorithm’s execution time.

Flexible operation of modular electrochemical CO2 reduction processes

Electrochemical CO2 reduction (eCO2R) is an emerging technology that is capable of producing various organic chemicals from CO2, but its high electricity cost is a big economic obstacle. One solution to reduce the cumulative electricity cost is demand side management, i.e., to adjust the power load based on time-variant electricity prices. However, varying the power load of CO2-electrolyzers often leads to changes in Faraday efficiency towards target components and thereby influences the product composition. Such deviations from the target product composition may be undesired for downstream processes. We tackle this challenge by proposing a flexible operating scheme for a modular eCO2R process. We formulate the economically optimal operation of an eCO2R process with multiple electrolyzer stacks as a parallel-machine scheduling problem. Adjusting the power load of each sub-process properly, we can save electricity costs while the desired product composition is met at any time. We apply an algorithm based on wavelet transform to solve the resulting large-scale nonlinear scheduling problem in tractable time. We solve each optimization problem with a deterministic global optimization software MAiNGO. We examine flexible operation of a modular eCO2R process for syngas production. The case studies show that the modular structure enables savings in the cumulative electricity cost of the eCO2R process via flexible operation while deviations in the syngas composition could be reduced. Also, the maximum ramping speed of the entire process is found to be a key parameter that strongly influences the cost saving.

Parallel Identical Machines Scheduling to Minimize the Maximum Inter-completion Time with Uncertain Processing Time

Motivated by parallel machines scheduling in practice that receive unplanned urgent job under uncertain environment, and it usually requires a response as soon as possible since its high-priority. This paper considers the response time to urgent job in a worst-case as an evaluation indicator. To the best of our knowledge, this paper is the first to study a parallel identical machine scheduling problem, which is to minimize the largest waiting time of an urgent job with the uncertain processing time of regular jobs. The objective of this problem is depicted to minimize the inter-completion time, that is, the maximum difference of the completion times between any two consecutively completed jobs. We first establish a stochastic programming model, and propose a scenario-reduction based sample average approximation method to solve this uncertain problem.

A Bi-Objective Optimization Approach for a Parallel Machine Scheduling Problem Minimizing Total Quality Loss and Total Tardiness

A Bi-Objective Optimization Approach for a Parallel Machine Scheduling Problem Minimizing Total Quality Loss and Total Tardiness

A mixed integer formulation and an efficient metaheuristic for the unrelated parallel machine scheduling problem: Total tardiness minimization

In this paper, the unrelated parallel machine scheduling problem with the objective of minimizing the total tardiness is addressed. For such a problem, a mixed-integer linear programming (MILP) formulation, that considers assignment and positional variables, is presented. In addition, an iterated local search (ILS) algorithm that produces high-quality solutions in reasonable times is proposed for higher size instances. The ILS robustness was determined by comparing its performance with the results provided by the MILP. The instances used in this paper were constructed under a new approach which results in tighter due dates than the previous generation method for this problem. The proposed MILP formulation was able to solve instances of up to 150 jobs and 20 machines. Regarding the ILS, it yielded high-quality solutions in a reasonable time, solving instances of a size up to 400 jobs and 20 machines. Experimental results confirm that both approaches are efficient and promising. • A new mixed integer programming formulation for the Unrelated Parallel Machine Scheduling Problem. • The formulation uses positional variables, is linear and does not use the Big-M technique. • The formulation solves instances up to 150 jobs and 20 machines to optimality. • An efficient iterated local search metaheuristic is presented. • A potential issue in the commonly used instance-generation procedure is discussed and a new set of instances is proposed.

Bi-objective optimization of identical parallel machine scheduling with flexible maintenance and job release times

This paper investigates an identical parallel machine scheduling problem with flexible maintenance and job release times and attempts to optimize two objectives: the minimization of the makespan and total tardiness simultaneously. A mixed-integer programming (MIP) model for solving small-scale instances is presented first, and then a modified NSGA-Ⅱ (M-NSGA-Ⅱ) algorithm is constructed for solving medium- and large-scale instances by incorporating several strategies. These strategies include: (ⅰ) the proposal of a decoding method based on dynamic programming, (ⅱ) the design of dynamic probability crossover and mutation operators, and (ⅲ) the presentation of neighborhood search method. The parameters of the proposed algorithm are optimized by the Taguchi method. Three scales of problems, including 52 instances, are generated to compare the performance of different optimization methods. The computational results demonstrate that the M-NSGA-Ⅱ algorithm obviously outperforms the original NSGA-II algorithm when solving medium- and large-scale instances, although the time taken to solve the instances is slightly longer.

Approximate dynamic programming for an energy-efficient parallel machine scheduling problem

In this paper, we propose an approximate dynamic programming approach for an energy-efficient unrelated parallel machine scheduling problem. In this scheduling problem, jobs arrive at the system randomly, and each job’s ready and processing times become available when an order is placed. Therefore, we consider the online version of the problem. Our objective is to minimize a combination of makespan and the total energy costs. The energy costs include cost of energy consumption of machines for switching on, processing, and idleness. We propose a binary program to solve the optimization problem at each stage of the approximate dynamic program. We compare the results of the approximate programming approach against an integer linear programming formulation of the offline version of the scheduling problem and an existing heuristic method suitable for scheduling problem with ready times. The results show that the approximate dynamic programming algorithm outperforms the two off-line methods in terms of solution quality and computational time.

Novel Formulations and Logic-Based Benders Decomposition for the Integrated Parallel Machine Scheduling and Location Problem

We investigate the discrete parallel machine scheduling and location problem, which consists of locating multiple machines to a set of candidate locations, assigning jobs from different locations to the located machines, and sequencing the assigned jobs. The objective is to minimize the maximum completion time of all jobs, that is, the makespan. Though the problem is of theoretical significance with a wide range of practical applications, it has not been well studied as reported in the literature. For this problem, we first propose three new mixed-integer linear programs that outperform state-of-the-art formulations. Then, we develop a new logic-based Benders decomposition algorithm for practical-sized instances, which splits the problem into a master problem that determines machine locations and job assignments to machines and a subproblem that sequences jobs on each machine. The master problem is solved by a branch-and-cut procedure that operates on a single search tree. Once an incumbent solution to the master problem is found, the subproblem is solved to generate cuts that are dynamically added to the master problem. A generic no-good cut is first proposed, which is later improved by some strengthening techniques. Two optimality cuts are also developed based on optimality conditions of the subproblem and improved by strengthening techniques. Numerical results on small-sized instances show that the proposed formulations outperform state-of-the-art ones. Computational results on 1,400 benchmark instances with up to 300 jobs, 50 machines, and 300 locations demonstrate the effectiveness and efficiency of the algorithm compared with current approaches. Summary of Contribution: This paper employs operations research methods and computing techniques to address an NP-hard combinatorial optimization problem: the parallel discrete machine scheduling and location problem. The problem is of practical significance but has not been well studied in the literature. For the problem, we formulate three novel mixed-integer linear programs that outperform state-of-the-art formulations and develop a new logic-based Benders decomposition algorithm. Extensive computational experiments on 1,400 benchmark instances with up to 300 jobs, 50 machines, and 300 locations are conducted to evaluate the performance of the proposed models and algorithms.

Decomposition approaches for parallel machine scheduling of step-deteriorating jobs to minimize total tardiness and energy consumption

In this paper, an identical parallel machine scheduling problem with step-deteriorating jobs is considered to minimize the weighted sum of tardiness cost and extra energy consumption cost. In particular, the actual processing time of a job is assumed to be a step function of its starting time and its deteriorating threshold. When the starting time of a job is later than its deteriorating threshold, the job faces two choices: (1) maintaining its status in holding equipment and being processed with a base processing time and (2) consuming an extra penalty time to finish its processing. The two work patterns need different amounts of energy consumption. To implement energy-efficient scheduling, the selection of the pre-processing patterns must be carefully considered. In this paper, a mixed integer linear programming (MILP) model is proposed to minimize the total tardiness cost and the extra energy cost. Decomposition approaches based on logic-based Benders decomposition (LBBD) are developed by reformulating the studied problem into a master problem and some independent sub-problems. The master problem is relaxed by only making assignment decisions. The sub-problems are to find optimal schedules in the job-to-machine assignments given by the master problem. Moreover, MILP and heuristic based on Tabu search are used to solve the sub-problems. To evaluate the performance of our methods, three groups of test instances were generated inspired by both real-world applications and benchmarks from the literature. The computational results demonstrate that the proposed decomposition approaches can compute competitive schedules for medium- and large-size problems in terms of solution quality. In particular, the LBBD with Tabu search performs the best among the suggested four methods.

Outranking-based multi-objective PSO for scheduling unrelated parallel machines with a freight industry-oriented application

This paper presents Outranking-based Particle Swarm Optimisation (O-PSO) a novel metaheuristic to address the multi-objective Unrelated Parallel Machine Scheduling Problem. It is a particle swarm optimisation algorithm enriched with the preferences of the Decision Maker (DM), articulated in a fuzzy relational system based on ELECTRE III. Unlike other multi-objective metaheuristics, O-PSO searches for the Region of Interest (RoI) instead of approximating a sample of the complete Pareto frontier. The RoI is the subset consisting of those Pareto-efficient solutions that satisfy the outranking relations, that is, they are the best solutions in terms of the DM’s system of preferences. Therefore, O-PSO not only approximates the Pareto solutions but also supports multicriteria decision analysis of the schedules. The efficiency of O-PSO is validated on a benchmark of synthetic instances from the scientific literature, where the Wilcoxon rank-sum test provides statistical evidence that O-PSO offers high-quality solutions when compared with two state-of-the-art metaheuristics; specifically, O-PSO is capable of generating a greater proportion of solutions (on average, ranging from 7% to 14%) dominating those of the state-of-the-art algorithms, as well as finding more solutions (from 13% to 18%) that satisfy the DM’s preferences. O-PSO is also applied to a real-world case study in the transport industry to provide evidence for its applicability.

Exact and metaheuristic approaches for unrelated parallel machine scheduling

In this paper, we study an important real-life scheduling problem that can be formulated as an unrelated parallel machine scheduling problem with sequence-dependent setup times, due dates, and machine eligibility constraints. The objective is to minimise total tardiness and makespan. We adapt and extend a mathematical model to find optimal solutions for small instances. Additionally, we propose several variants of simulated annealing to solve very large-scale instances as they appear in practice. We utilise several different search neighbourhoods and additionally investigate the use of innovative heuristic move selection strategies. Further, we provide a set of real-life problem instances as well as a random instance generator that we use to generate a large number of test instances. We perform a thorough evaluation of the proposed techniques and analyse their performance. We also apply our metaheuristics to approach a similar problem from the literature. Experimental results show that our methods are able to improve the results produced with state-of-the-art approaches for a large number of instances.

Application of a hybrid evolutionary algorithm to resource-constrained parallel machine scheduling with setup times

We address the problem of resource-constrained parallel machine scheduling with setup times in the practical context of microelectronic components manufacturing. This NP-hard problem is addressed using a biased random-key genetic algorithm hybridized with tailored local search procedures organized using variable neighborhood descent. The only benchmark available in the literature is utilized, and the optimal results are presented for all instances. Two new sets with 270 challenging instances are proposed to assess the quality of the solutions reported by the proposed method. A series of experiments are conducted to generate lower and upper bounds, including four models, two list processing heuristics from the literature and an implementation of a general variable neighborhood search metaheuristic. The bounds are used as reference values. The average percentage distances from the lower and upper bounds were 22.44% and −7.62%, respectively.

Branch-and-price approach for robust parallel machine scheduling with sequence-dependent setup times

This paper studies a machine scheduling problem that minimizes the worst-case total tardiness for unrelated parallel machines with sequence-dependent setup and uncertain processing times. We propose a robust optimization reformulation of the related machine scheduling problem and discuss several important properties of the mathematical model and the reformulation approach. The proposed model generalizes robust parallel machine scheduling problems by including sequence-dependent setup times and ellipsoidal uncertainty sets. Another key contribution of the paper is to show that scheduling problems usually have alternative optimal solutions for the worst-case tardiness objective, whose performance under nominal processing times may vary or vice a versa. This issue has been addressed by studying the Pareto efficient extensions of the proposed robust optimization models to provide solutions that are immune to changes in processing times. A branch-and-price algorithm has been developed to solve realistically sized instances in less than one hour, which a commercial solver cannot achieve. Numerical results show the effectiveness of the proposed approach since realistically sized instances such as (4 machines, 32 jobs) and (150 machines, 300 jobs) can be solved to optimality within the time limit, and the (average) objective function value improvement made by the robust approach can get as high as 56% compared with the (nominal) optimal solutions that ignore uncertainty in problem data.

Advanced discrete firefly algorithm with adaptive mutation‐based neighborhood search for scheduling unrelated parallel machines with sequence‐dependent setup times

The unrelated parallel machine scheduling problem with sequence-dependent setup times is addressed in this paper with the objective of minimizing the elapsed time between the start and finish of a sequence of operations in a set of unrelated machines. The machines are considered unrelated because the processing speed is dependent on the job being executed and not on the individual machines. Generally, the problem is considered NP-hard, as it presents additional complexity to find an optimal solution in terms of minimum makespan. An advanced firefly metaheuristic optimization algorithm is introduced to solve this problem. The proposed method, called the FAII algorithm, aims to improve the standard firefly algorithm's performance by incorporating an enhanced global best solution update mechanism and adaptive mutation-based local and global neighborhood search scheme to improve the quality of the proposed algorithm's generated solution. Several experiments were conducted to compare and validate the proposed algorithms' performance on small and large-scale benchmarked problem instances with up to 12 machines and 120 job combinations. Moreover, the performance of the FAII was also compared with eight other metaheuristic algorithms, which were implemented in parallel with the FAII method. Furthermore, the numerical results of the FAII algorithm were compared with the scheduling results of six other well-known metaheuristics from the literature. The comparison results backed with a comprehensive statistical analysis showed the superiority of the enhanced FA-style scheduling optimization over other metaheuristic methods to find good quality solutions or minimum average makespan.