Local search based methods for scheduling in the unrelated parallel machines environment
Introduction
Scheduling processes are important in many real-world situations, such as planning in manufacturing plants (Kofler et al., 2009, Ouelhadj and Petrovic, 2009), universities and schools (Lewis, Paechter, & Rossi-Doria, 2007), airports (Cheng et al., 1999, Hansen, 2004), hospitals (Burke et al., 2004, Petrovic and Castro, 2011), and many others. In such problems, the goal is to assign a certain number of jobs or tasks to a limited number of machines or resources in such a way that all required constraints are satisfied and a certain user-defined criterion is optimised (Pinedo, 2012). Solving most scheduling problems is difficult because they are NP hard. Exact methods can provide optimal solutions to such problems (Fanjul-Peyro, Ruiz, & Perea, 2019), but they cannot be applied to large-scale problems because it is not possible to enumerate the entire search space. Therefore, most research focused on the development and use of methods that do not necessarily provide optimal results, but can achieve good solutions in a reasonable amount of time. The studies mainly focused on two types of methods, approximation methods (Lenstra, Shmoys, & Tardos, 1990) and heuristic methods (Morton & Pentico, 1993). Approximation methods provide some guarantee of optimality of the solution (Pinedo, 2012, Wotzlaw, 2012), while heuristic methods provide no guarantees.
Of the previous two types of methods, heuristic algorithms have attracted considerable attention because they can be adapted to a wider range of problems and are easier to design. Heuristic methods used for solving scheduling problems can be divided into problem-specific heuristics and metaheuristics. Problem-specific heuristics usually appear in the form of dispatching rules (DRs) (Braun et al., 2001; Đurasević & Jakobović, 2018b; Maheswaran, Ali, Siegel, Hensgen, & Freund, 1999). DRs are simple heuristics that build the schedule iteratively by deciding which scheduling decision should be made next, i.e., which job should be scheduled on which machine. Because of this, they have limited visibility into the problem, which affects their performance. They can generate the schedule extremely fast, which makes them ideal for dynamic scheduling problems. On the other hand, metaheuristic algorithms search as much of the solution space as possible by starting with complete solutions and iteratively improving them by introducing changes (Hart, Ross, & Corne, 2005). Such methods can achieve extremely good results because they do not construct the schedule greedily, but rather try to improve the existing ones.
Various metaheuristics have been applied to the problem of scheduling on parallel unrelated machines. Researchers mainly used genetic algorithms (GAs) (Holland, 1992), tabu search (TS) (Glover, 1990), simulated annealing (SA) (Kirkpatrick, Gelatt, & Vecchi, 1983), ant colony optimisation (ACO), variable neighbourhood search (VNS), variable neighbourhood descent (VND) (Mladenović & Hansen, 1997), greedy randomised adaptive search procedure (GRASP) (Feo & Resende, 1995), and others. Among such a large number of methods, it is difficult to choose the one that is most appropriate for the problem under consideration. Moreover, the above algorithms are often combined into sophisticated methods that are difficult to understand and reproduce. In many cases, novel metaheuristics are also proposed to deal with scheduling problems. However, such methods usually offer limited novelty (Sörensen, 2015) or the problem in question could have been solved more effectively with a simpler method that yields competitive results (Molnar, Jakobović, & Pavelić, 2016).
Local search (LS) operators are popular methods for searching in the neighbourhood of the current solution. They have been used in various optimisation problems, including unrelated machines. Due to their increasing popularity, a wide range of LS operators have been proposed in the literature, along with metaheuristics that define how they should be applied. In addition, hybrid algorithms are proposed to further improve performance, which increases their complexity. This inevitably leads to the situation where it is difficult to choose the right LS operators or to determine which strategies should be used to apply them to a problem. However, instead of developing new LS operators or hybrid methods, one should consider if with already existing operators it is possible to construct simple methods that produce high-quality results.
This paper investigates whether various LS operators can be combined into simple but efficient methods for solving the problem of scheduling parallel unrelated machines. For this purpose, different approaches from the literature were analysed, based on which the most important parts of LS methods were identified, such as initial solution generation, perturbation and LS operators. Based on these observations, two simple iterative LS methods were defined. To provide an exhaustive analysis of the proposed methods, several LS operators were selected from the literature and used as building blocks for the proposed methods. The studied methods were compared with several metaheuristics and the results showed that the iterative LS procedures easily outperform all of the methods. These results indicate that simple LS based methods are as powerful as the more complex metaheuristics. This conclusion is important because it shows that it is more beneficial to focus on constructing simple LS based methods than to apply metaheuristics that may not be appropriate for the problem. The contributions of this work can be summarised as follows:
- 1.
Definition of two simple iterative LS methods and identification of the key design decisions in their development.
- 2.
Analyse the effectiveness of the various LS operators proposed in the literature.
- 3.
Investigation of the influence of different design decisions (initial solution generation, solution acceptance criterion, application of path relinking) on the performance of the simple iterative LS methods.
- 4.
Analysis of the performance of the simple iterative local search methods compared to several previously applied metaheuristics.
The paper is organised as follows. The literature review of heuristic methods applied to the unrelated machines environment problem is given in Section 2. Section 3 contains the description of the unrelated machines scheduling environment. The overview of the methods applied to solve the above problem is given in Section 4. Section 5 describes the experimental setup and outlines the obtained results. Finally, Section 6 concludes the paper and provides guidelines for future research.
Section snippets
Literature overview
Until now, a large number of studies have addressed the application of metaheuristics to the unrelated machines scheduling problem. Glass, Potts, and Shade (1994) applied GA, SA, and TS to optimise the makespan criterion. SA and TS used two neighbourhood operators, job reassignment and job interchange. The reassignment neighbourhood moved an job from one machine to another, while the interchange neighbourhood swapped two jobs on different machines. The results show that all three algorithms
Problem definition
The scheduling problem considered in this paper can be classified as (Pinedo, 2012), which means that the considered environment is the parallel unrelated machines environment with job release times and the total weighted tardiness criterion is optimised. A single problem instance in this environment consists of jobs, where each job must be scheduled on one of the available machines. It is assumed that the number of jobs and machines is finite. The subscript refers to jobs, while
Iterated local search procedures
Over the years, several LS based methods have been proposed, such as GRASP (Feo & Resende, 1995), VNS (Hansen and Mladenovic, 2001, Mladenović and Hansen, 1997), and VND (Hansen, Mladenovic, Todosijević, & Hanafi, 2016). These methods work in similar ways, with the differences lying in details such as whether or not one or more LS operators are used, or whether or not the LS operators are applied in a deterministic manner. None of these procedures are explicitly used in this work, instead a
Experimental setup
To test the considered algorithms, a set of problem instances was generated using procedures from the literature (Kim et al., 2003, Lee et al., 2013, Lin et al., 2013, Vlašić et al., 2020). The generated set consists of 60 problem instances with different properties. The instances were generated with different combinations of the number of jobs and machines, with the number of jobs set to 12, 25, 50, and 100 and the number of machines set to 3, 6, and 10. Job processing times were generated
Conclusion
This paper investigated the performance of various metaheuristics for the parallel unrelated machines scheduling problem. The goal was to find out whether it is necessary to use complicated metaheuristics for the above problem, or whether it is possible to obtain satisfactory results using simpler methods. The experiments show that simple methods based on LS operators are more powerful than some complex or hybrid methods. Metaheuristics such as GA, ACO, and TS were not able to give as good
CRediT authorship contribution statement
Lucija Ulaga: Conceptualization, Software, Validation, Investigation, Visualization. Marko Đurasević: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing – original draft, Writing – review & editing, Visualization, Supervision. Domagoj Jakobović: Conceptualization, Validation, Writing – original draft, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
This work has been supported in part by Croatian Science Foundation, Republic of Croatia under the project IP-2019-04-4333.
References (67)
Non-identical parallel machine scheduling using genetic algorithm
Expert Systems with Applications
(2011)- et al.
Parallel-machine scheduling problems with sequence-dependent setup times using an ACO, SA and VNS hybrid algorithm
Expert Systems with Applications
(2009) - et al.
A comparison of eleven static heuristics for mapping a class of independent tasks onto heterogeneous distributed computing systems
Journal of Parallel and Distributed Computing
(2001) - et al.
Integrating dominance properties with genetic algorithms for parallel machine scheduling problems with setup times
Applied Soft Computing
(2011) - et al.
Hybrid GRASP heuristics to solve an unrelated parallel machine scheduling problem with earliness and tardiness penalties
Electronic Notes in Theoretical Computer Science
(2014) Enhanced symbiotic organisms search algorithm for unrelated parallel machines manufacturing scheduling with setup times
Knowledge-Based Systems
(2019)- et al.
Iterated greedy local search methods for unrelated parallel machine scheduling
European Journal of Operational Research
(2010) - et al.
Size-reduction heuristics for the unrelated parallel machines scheduling problem
Computers & Operations Research
(2011) - et al.
Scheduling unrelated parallel machines with optional machines and jobs selection
Computers & Operations Research
(2012) - et al.
Reformulations and an exact algorithm for unrelated parallel machine scheduling problems with setup times
Computers & Operations Research
(2019)
Unrelated parallel machine scheduling using local search
Mathematical and Computer Modelling
Genetic search methods in air traffic control
Computers & Operations Research
Variable neighborhood search: Principles and applications
European Journal of Operational Research
Rule-based meta-heuristics for integrated scheduling of unrelated parallel machines, batches, and heterogeneous delivery trucks
Applied Soft Computing
Unrelated parallel machine scheduling with setup times using simulated annealing
Robotics and Computer-Integrated Manufacturing
Unrelated parallel machine scheduling with setup times and a total weighted tardiness objective
Robotics and Computer-Integrated Manufacturing
A comparison of five hybrid metaheuristic algorithms for unrelated parallel-machine scheduling and inbound trucks sequencing in multi-door cross docking systems
Applied Soft Computing
Scheduling unrelated parallel machines with sequence-dependent setups
Computers & Operations Research
Dynamic mapping of a class of independent tasks onto heterogeneous computing systems
Journal of Parallel and Distributed Computing
Variable neighborhood search
Computers & Operations Research
Robust meta-heuristics implementation for unrelated parallel machines scheduling problem with rework processes and machine eligibility restrictions
Computers & Industrial Engineering
A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem
European Journal of Operational Research
Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems
Computers & Industrial Engineering
A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times
European Journal of Operational Research
Improving genetic algorithm performance by population initialisation with dispatching rules
Computers & Industrial Engineering
A comparative study of solution representations for the unrelated machines environment
Computers & Operations Research
Design of high-performing hybrid meta-heuristics for unrelated parallel machine scheduling with machine eligibility and precedence constraints
Engineering Optimization
Optimize unrelated parallel machines scheduling problems with multiple limited additional resources, sequence-dependent setup times and release date constraints
IEEE Access
Comparisons of metaheuristic algorithms for unrelated parallel machine weighted earliness/tardiness scheduling problems
Evolutionary Intelligence
A two-stage ant colony optimization algorithm to minimize the makespan on unrelated parallel machines with sequence-dependent setup times
Journal of Intelligent Manufacturing
Genetic algorithms and random keys for sequencing and optimization
ORSA Journal on Computing
The state of the art of nurse rostering
Journal of Scheduling
Air traffic control using genetic search techniques
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