Review on unrelated parallel machine scheduling problem with additional resources

This study extends the best hybrid ant colony optimization variant developed by Liao et al. (2014) for crisp unrelated parallel machine scheduling problems to solve fuzzy unrelated parallel machine scheduling problems in consideration of trapezoidal fuzzy processing times, trapezoidal fuzzy sequencing dependent setup times and trapezoidal fuzzy release times. The objective is to find the best schedule taking minimum fuzzy makespan in completing all jobs. In this study, fuzzy arithmetic is used to determine fuzzy completion times of jobs and defuzzification function is used to convert fuzzy numbers back to crisp numbers for ranking. Eight fuzzy ranking methods are tested to find the most feasible one to be employed in this study. The fuzzy arithmetic testing includes four different cases and each case with the following operations separately, i.e., addition, subtraction, multiplication and division, to investigate the spread of fuzziness as fuzzy numbers are subject to more and more number of operations. The effect of fuzzy ranking methods on hybrid ant colony optimization (hACO) is investigated. To prove the correctness of our methodology and coding, unrelated parallel machine scheduling with fuzzy numbers and crisp numbers are compared based on scheduling problems up to 15 machines and 200 jobs. Relative percentage deviation (RPD) is used to evaluate the performance of hACO in solving fuzzy unrelated parallel machine scheduling problems. A numerical study on large-scale scheduling problems up to 20 machines and 200 jobs is conducted to assess the performance of the hACO algorithm. For comparison, a discrete particle swarm optimization (dPSO) algorithm is implemented for fuzzy unrelated parallel machine scheduling problem as well. The results show that the hACO has better performance than dPSO not only in solution quality in terms of RPD value, but also in computational time.

This paper addresses the problem of makespan minimization on unrelated parallel machines with sequence dependent setup times. The symbiotic organisms search (SOS) algorithm is a new and popular global optimization technique that has received wide acceptance in recent years from researchers in continuous and discrete optimization domains. An improved SOS algorithm is developed to solve the parallel machine scheduling problem. Since the standard SOS algorithm was originally developed to solve continuous optimization problems, a new solution representation and decoding procedure is designed to make the SOS algorithm suitable for the unrelated parallel machine scheduling problem (UPMSP). Similarly, to enhance the solution quality of the SOS algorithm, an iterated local search strategy based on combining variable numbers of insertion and swap moves is incorporated into the SOS algorithm. More so, to further improve the SOS optimization speed and performance, the longest processing time first (LPT) rule is used to design a machine assignment heuristic that assigns processing machines to jobs based on the machine dynamic load-balancing mechanism. Subsequently, the machine assignment scheme is incorporated into SOS algorithms and used to solve the UPMSP. The performances of the proposed methods are evaluated by comparing their solutions with other existing techniques from the literature. A number of statistical tests were also conducted to determine the variations in performance for each of the techniques. The experimental results showed that the SOS with LPT (SOS-LPT) heuristic has the best performance compared to other tested method, which is closely followed by SOS algorithm, indicating that the two proposed algorithms’ solution approaches are reasonable and effective for solving large-scale UPMSPs.

This article addresses a challenging industrial problem known as the unrelated parallel machine scheduling problem (UPMSP) with sequence-dependent setup times. In UPMSP, we have a set of machines and a group of jobs. The goal is to find the optimal way to schedule jobs for execution by one of the several available machines. UPMSP has been classified as an NP-hard optimisation problem and, thus, cannot be solved by exact methods. Meta-heuristic algorithms are commonly used to find sub-optimal solutions. However, large-scale UPMSP instances pose a significant challenge to meta-heuristic algorithms. To effectively solve a large-scale UPMSP, this article introduces a two-stage evolutionary variable neighbourhood search (EVNS) methodology. The proposed EVNS integrates a variable neighbourhood search algorithm and an evolutionary descent framework in an adaptive manner. The proposed evolutionary framework is employed in the first stage. It uses a mix of crossover and mutation operators to generate diverse solutions. In the second stage, we propose an adaptive variable neighbourhood search to exploit the area around the solutions generated in the first stage. A dynamic strategy is developed to determine the switching time between these two stages. To guide the search towards promising areas, a diversity-based fitness function is proposed to explore different locations in the search landscape. We demonstrate the competitiveness of the proposed EVNS by presenting the computational results and comparisons on the 1640 UPMSP benchmark instances, which have been commonly used in the literature. The experiment results show that our EVNS obtains better results than the compared algorithms on several UPMSP instances.

This research proposes a two-phase heuristic and an improvement procedure for scheduling unrelated parallel machines problems to minimize makespan.

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.

Heuristic algorithms for the unrelated parallel machine scheduling problem with one scarce additional resource

Unrelated parallel machine scheduling problem with special controllable processing times and setups

In this paper, an unrelated parallel machine scheduling problem with sequence and machine-dependent setup times and makespan minimization is studied. A new makespan linearization and several mixed integer formulations are proposed for this problem. These formulations outperform the previously published formulations regarding size of instances and computational time to reach optimal solutions. Using these models, it is possible to solve instances six times larger than what was previously solved and to obtain optimal solutions on instances of the same size up to four orders of magnitude faster. A metaheuristic algorithm based on multi-start algorithm and variable neighbourhood descent metaheuristic is proposed. Composite movements were defined for the improvement phase of the proposed metaheuristic algorithm that considerably improved the performance of the algorithm providing small deviations from optimal solutions in medium-sized instances and outperforming the best known solutions for large instances.

PurposeThe purpose of this paper is to present a hybrid meta-heuristic based on genetic algorithms (GAs), simulated annealing, variable neighborhood descent and path relinking for solving the variant of the unrelated parallel machine scheduling problem considering sequence-dependent setup times.Design/methodology/approachThe authors carried out computational experiments on literature problem instances proposed by Vallada and Ruiz (2011) and Arnaout et al. (2010) to test the performance of the proposed meta-heuristic. The objective function adopted was makespan minimization, and the authors used relative deviation, average and population standard deviation as performance criteria.FindingsThe results indicate the competitivity of the proposed approach and its superiority in comparison with several other algorithms. In small instances proposed by Vallada and Ruiz (2011) and on small and large instances proposed by Arnaout et al. (2010), the proposed approach presented the best results in most tested problem instances.Practical implicationsIn small instances proposed by Vallada and Ruiz (2011) and on small and large instances proposed by Arnaout et al. (2010), the proposed approach presented the best results in most tested problem instances.Originality/valueThe proposed approach presented high-quality results, with an innovative hybridization of a GA and neighborhood search algorithms, tested in diverse instances of literature. Furthermore, the case study demonstrated that the proposed approach is recommended for solving real-world problems.

Reformulations and an exact algorithm for unrelated parallel machine scheduling problems with setup times

Multiple-objective heuristics for scheduling unrelated parallel machines

Bicriteria scheduling problem for unrelated parallel machines with release dates

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.

Hybrid tabu search and a truncated branch-and-bound for the unrelated parallel machine scheduling problem

The Unrelated Parallel Machines Scheduling Problem (UPMSP) with sequence-dependent setup times has been widely applied to cloud computing, edge computing and so on. When the setup times are ignored, UPMSP will be a NP problem. Moreover, when considering the sequence related setup times, UPMSP is difficult to solve, and this situation will be more serious in the case of high-dimensional. This work firstly select the maximum completion time as the optimization objective, which establishes a mathematical model of UPMSP with sequence-dependent setup times. In addition, an improved firefly algorithm with courtship learning is proposed. Finally, in order to provide an approximate solution in an acceptable time, the proposed algorithm is applied to solve the UPMSP with sequence-dependent setup times. The experimental results show that the proposed algorithm has competitive performance when dealing with UPMSP with sequence-dependent setup times.